Podcast
Questions and Answers
The sum of 2 rational numbers is always what type of number?
The sum of 2 rational numbers is always what type of number?
rational
The product of 2 rational numbers is always what type of number?
The product of 2 rational numbers is always what type of number?
rational
The sum of an irrational and a rational number is always what type of number?
The sum of an irrational and a rational number is always what type of number?
irrational
The product of an irrational and a rational number is always what type of number?
The product of an irrational and a rational number is always what type of number?
What is the name for the specific number system that encompasses both rational and irrational numbers?
What is the name for the specific number system that encompasses both rational and irrational numbers?
What does the symbol 'R' with a line through it represent?
What does the symbol 'R' with a line through it represent?
What is a set, in mathematical terms?
What is a set, in mathematical terms?
What is an element of a set?
What is an element of a set?
What symbols are used to list sets?
What symbols are used to list sets?
What is a subset of a set?
What is a subset of a set?
What type of number is the sum of two rational numbers?
What type of number is the sum of two rational numbers?
What type of number is the product of two rational numbers?
What type of number is the product of two rational numbers?
What type of number is the sum of a rational number and an irrational number?
What type of number is the sum of a rational number and an irrational number?
What type of number is the product of a rational number and an irrational number?
What type of number is the product of a rational number and an irrational number?
Which of the following number sets does 0 belong to?
Which of the following number sets does 0 belong to?
What is a defining characteristic of an irrational number?
What is a defining characteristic of an irrational number?
What are some examples of irrational numbers?
What are some examples of irrational numbers?
In consecutive integer problems, what can 'x' represent?
In consecutive integer problems, what can 'x' represent?
If you need to use specifically even or odd integers, what adjustments do you make to 'x' when representing consecutive integers?
If you need to use specifically even or odd integers, what adjustments do you make to 'x' when representing consecutive integers?
What are consecutive integers?
What are consecutive integers?
In word problems, 'x' can represent the value of what?
In word problems, 'x' can represent the value of what?
How can the difference of two rational numbers be rewritten?
How can the difference of two rational numbers be rewritten?
When simplifying non-perfect square roots, what factor should you keep?
When simplifying non-perfect square roots, what factor should you keep?
What type of number is the difference of two rational numbers?
What type of number is the difference of two rational numbers?
What type of number is the difference of a rational number and an irrational number?
What type of number is the difference of a rational number and an irrational number?
What is an identity?
What is an identity?
What is the solution set for an identity?
What is the solution set for an identity?
What is a contradiction?
What is a contradiction?
What does a literal equation contain?
What does a literal equation contain?
Describe what a formula is.
Describe what a formula is.
What happens when you divide an inequality by a negative number?
What happens when you divide an inequality by a negative number?
What is the recommended approach to solving a literal equation when your variable is in the denominator and the numerator has another variable?
What is the recommended approach to solving a literal equation when your variable is in the denominator and the numerator has another variable?
When you multiply or divide an equation by a value, what needs to be done to ensure accuracy?
When you multiply or divide an equation by a value, what needs to be done to ensure accuracy?
What is the best strategy to use with literal equations when possible?
What is the best strategy to use with literal equations when possible?
Under what circumstance does the direction of the inequality symbol get reversed in an inequality?
Under what circumstance does the direction of the inequality symbol get reversed in an inequality?
What does the solution being all real numbers imply in an equation, when variables cancel out and you are left with a true statement?
What does the solution being all real numbers imply in an equation, when variables cancel out and you are left with a true statement?
What does it indicate when variables cancel out in an equation, resulting in a false statement?
What does it indicate when variables cancel out in an equation, resulting in a false statement?
Explain what kind of solutions does the inequality x < -2 have?
Explain what kind of solutions does the inequality x < -2 have?
Where is it ideal to have the variable x (or any variable) positioned in an equation?
Where is it ideal to have the variable x (or any variable) positioned in an equation?
Why are all values of x true in an equality?
Why are all values of x true in an equality?
What is the general rule for rounding answers in word problems, keeping in mind the context of the problem?
What is the general rule for rounding answers in word problems, keeping in mind the context of the problem?
What is a crucial step after solving a mathematical problem?
What is a crucial step after solving a mathematical problem?
What is a compound inequality?
What is a compound inequality?
What is the first step in solving a compound inequality?
What is the first step in solving a compound inequality?
What is the second step in solving a compound inequality?
What is the second step in solving a compound inequality?
Describe the process of setting up a notation for the solution of a compound inequality.
Describe the process of setting up a notation for the solution of a compound inequality.
In a compound inequality with AND, what does the intersection represent on the graph?
In a compound inequality with AND, what does the intersection represent on the graph?
How should you handle the 'and' or 'or' in a compound inequality throughout the solving process?
How should you handle the 'and' or 'or' in a compound inequality throughout the solving process?
If there is no intersection in the graph of an AND compound inequality, what does that indicate about the solution?
If there is no intersection in the graph of an AND compound inequality, what does that indicate about the solution?
What does the solution of OR compound inequalities represent on the graph?
What does the solution of OR compound inequalities represent on the graph?
Describe the union as it pertains to compound inequalities.
Describe the union as it pertains to compound inequalities.
If a compound inequality with OR has arrows pointing throughout the entire number line, what does this mean about the solution set?
If a compound inequality with OR has arrows pointing throughout the entire number line, what does this mean about the solution set?
What does a compound inequality combine?
What does a compound inequality combine?
What are the two types of compound inequalities?
What are the two types of compound inequalities?
If an AND compound inequality results in an OR graph, what is the implication?
If an AND compound inequality results in an OR graph, what is the implication?
If an OR compound inequality results in an AND graph, what does that mean?
If an OR compound inequality results in an AND graph, what does that mean?
In which type of compound inequality should you always isolate x (or the variable) in the middle?
In which type of compound inequality should you always isolate x (or the variable) in the middle?
When graphing a regular AND compound inequality, which areas should be shaded?
When graphing a regular AND compound inequality, which areas should be shaded?
In an AND compound inequality like 4 > x > -2, how do you approach the number arrangement?
In an AND compound inequality like 4 > x > -2, how do you approach the number arrangement?
What can you do to ensure the smallest number comes first in an AND compound inequality?
What can you do to ensure the smallest number comes first in an AND compound inequality?
In Absolute Value Equations/Inequalities, which symbol means AND?
In Absolute Value Equations/Inequalities, which symbol means AND?
How would you remember that Less than (regarding the variable) means AND and Greater than (regarding the variable) means OR?
How would you remember that Less than (regarding the variable) means AND and Greater than (regarding the variable) means OR?
What are the absolute value symbols?
What are the absolute value symbols?
You know you solved an absolute value inequality correctly if the graph works for the original and two other inequalities.
You know you solved an absolute value inequality correctly if the graph works for the original and two other inequalities.
What is the first step to solve absolute value inequalities/equations word problems?
What is the first step to solve absolute value inequalities/equations word problems?
What is the second step to solve absolute value inequalities/equations word problems?
What is the second step to solve absolute value inequalities/equations word problems?
Which steps are necessary to solve a regular absolute value inequality/equation without a word problem?
Which steps are necessary to solve a regular absolute value inequality/equation without a word problem?
To solve an equation in the form absolute value of A = b, where A represents a variable expression and b > or equal to 0 (basically positive), solve A = b AND A = -b.
To solve an equation in the form absolute value of A = b, where A represents a variable expression and b > or equal to 0 (basically positive), solve A = b AND A = -b.
What does absolute value mean?
What does absolute value mean?
What are the bars indicating absolute value called?
What are the bars indicating absolute value called?
What is the first step to solving any equation/inequality that involves absolute value?
What is the first step to solving any equation/inequality that involves absolute value?
What do you do after isolating the absolute value expression in an equation or inequality?
What do you do after isolating the absolute value expression in an equation or inequality?
What does the equation the absolute value of 2x - 5 = 13 mean?
What does the equation the absolute value of 2x - 5 = 13 mean?
How many points are 13 units away from zero? What are they?
How many points are 13 units away from zero? What are they?
What do you do to solve for x in the equation the absolute value of 2x - 5 = 13?
What do you do to solve for x in the equation the absolute value of 2x - 5 = 13?
When do you split the absolute value equation up into two separate ones?
When do you split the absolute value equation up into two separate ones?
How would you represent that both positive and negative values of a number work for the variable?
How would you represent that both positive and negative values of a number work for the variable?
In the broken up equations, what is the first equation?
In the broken up equations, what is the first equation?
In the broken up equations, what is the second equation?
In the broken up equations, what is the second equation?
Any equation in which the absolute value expression equals a negative value has no solution.
Any equation in which the absolute value expression equals a negative value has no solution.
Why does an absolute value equation have no solution when it equals a negative value?
Why does an absolute value equation have no solution when it equals a negative value?
0 is neither positive or negative, it's zero. There is no opposite of 0, therefore if you get something like the absolute value of q = 0, there is one solution.
0 is neither positive or negative, it's zero. There is no opposite of 0, therefore if you get something like the absolute value of q = 0, there is one solution.
What do Absolute Value Inequalities turn into?
What do Absolute Value Inequalities turn into?
How should you write the solution of an AND absolute value inequality?
How should you write the solution of an AND absolute value inequality?
Once you isolate the absolute value inequality, what is the next step?
Once you isolate the absolute value inequality, what is the next step?
To solve an inequality in the form absolute value of A < (or less than or equal to) b, where A is a variable expression and b > 0 , what is the next step?
To solve an inequality in the form absolute value of A < (or less than or equal to) b, where A is a variable expression and b > 0 , what is the next step?
To solve an inequality in the form absolute value of A > (or greater than or equal to) b, where A is a variable expression and b > 0, what is the next step?
To solve an inequality in the form absolute value of A > (or greater than or equal to) b, where A is a variable expression and b > 0, what is the next step?
Irrational/Rational = Irrational
Irrational/Rational = Irrational
Could 4.137604827 be rational?
Could 4.137604827 be rational?
What do you have to do to solve consecutive integer problems?
What do you have to do to solve consecutive integer problems?
What is the solution to 0 > -25 considered?
What is the solution to 0 > -25 considered?
What is a good thing to do when solving absolute value equations/inequalities?
What is a good thing to do when solving absolute value equations/inequalities?
What is a good thing to keep in mind when solving absolute value equations/inequalities?
What is a good thing to keep in mind when solving absolute value equations/inequalities?
What is the most accurate result?
What is the most accurate result?
What would be the most appropriate solution?
What would be the most appropriate solution?
Be on the look out for what scenario?
Be on the look out for what scenario?
What should you do if you get something about x's distance from another number being a certain amount of units (absolute value).
What should you do if you get something about x's distance from another number being a certain amount of units (absolute value).
What is the formula to find the slope between two points (x1, y1) and (x2, y2)?
What is the formula to find the slope between two points (x1, y1) and (x2, y2)?
A slope of 0 indicates a vertical line.
A slope of 0 indicates a vertical line.
What does 'rise' refer to when calculating slope?
What does 'rise' refer to when calculating slope?
The formula for calculating slope is m = (y2 - y1) / (x2 - ______).
The formula for calculating slope is m = (y2 - y1) / (x2 - ______).
Match the following types of slope with their description:
Match the following types of slope with their description:
In the equation $y = mx + b$, what does the variable 'm' represent?
In the equation $y = mx + b$, what does the variable 'm' represent?
The y-intercept of a graph is the point at which the line crosses the x-axis.
The y-intercept of a graph is the point at which the line crosses the x-axis.
What is the slope-intercept form of the equation for a line that passes through the point (0, 1) and (2, 2)?
What is the slope-intercept form of the equation for a line that passes through the point (0, 1) and (2, 2)?
In the function $y = 3x + 4$, the y-intercept is ______.
In the function $y = 3x + 4$, the y-intercept is ______.
Match the following equations with their corresponding slopes:
Match the following equations with their corresponding slopes:
What is the point-slope form equation of a line with a slope of $\frac{1}{2}$ that passes through the point (3, -2)?
What is the point-slope form equation of a line with a slope of $\frac{1}{2}$ that passes through the point (3, -2)?
The equation y - 3 = 1(x + 1) represents a line with a slope of 1.
The equation y - 3 = 1(x + 1) represents a line with a slope of 1.
Write an equation in point-slope form for the line that goes through the points (2, -1) and (-3, 3).
Write an equation in point-slope form for the line that goes through the points (2, -1) and (-3, 3).
In point-slope form, the equation is expressed as y - ______ = m(x - ______).
In point-slope form, the equation is expressed as y - ______ = m(x - ______).
What is the point-slope form of a linear equation?
What is the point-slope form of a linear equation?
The slope-intercept form is derived from the point-slope form.
The slope-intercept form is derived from the point-slope form.
Write the equation of a line in point-slope form that has a slope of 1/2 and passes through the point (3, -2).
Write the equation of a line in point-slope form that has a slope of 1/2 and passes through the point (3, -2).
The graph of the equation y - 3 = 1(x + 1) will cross the y-axis at ______.
The graph of the equation y - 3 = 1(x + 1) will cross the y-axis at ______.
Match the following point-slope equations with their corresponding slopes:
Match the following point-slope equations with their corresponding slopes:
What is the standard form of a linear equation?
What is the standard form of a linear equation?
The y-intercept of a line is where it intersects the x-axis.
The y-intercept of a line is where it intersects the x-axis.
In the equation 3x + 4y = 12, what are the intercepts if x=0?
In the equation 3x + 4y = 12, what are the intercepts if x=0?
The x-intercept occurs when y equals ______.
The x-intercept occurs when y equals ______.
Match the following variables with their descriptions:
Match the following variables with their descriptions:
Which of the following slopes represent parallel lines?
Which of the following slopes represent parallel lines?
All vertical lines are considered parallel.
All vertical lines are considered parallel.
What is the slope of a line that is perpendicular to a line with a slope of 2?
What is the slope of a line that is perpendicular to a line with a slope of 2?
The equation of a line in slope-intercept form that passes through the point (8, 9) and is parallel to the line y = 3x - 2 is y = 3x + ___
The equation of a line in slope-intercept form that passes through the point (8, 9) and is parallel to the line y = 3x - 2 is y = 3x + ___
Match the following pairs of lines to their classification:
Match the following pairs of lines to their classification:
Flashcards
What is a set?
What is a set?
a collection of objects such as numbers
What is an element of a set?
What is an element of a set?
an object within that set
What do you use to list sets?
What do you use to list sets?
Braces, {}
A subset of a set
A subset of a set
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What is a subset of every set?
What is a subset of every set?
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rational + rational =
rational + rational =
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rational x rational =
rational x rational =
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rational + irrational =
rational + irrational =
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rational x irrational =
rational x irrational =
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irrational + irrational =
irrational + irrational =
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irrational x irrational =
irrational x irrational =
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0 is
0 is
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An irrational number....
An irrational number....
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examples of irrationals are
examples of irrationals are
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In Consecutive Integer Problems, x can be
In Consecutive Integer Problems, x can be
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If you have to use specifically even or odd integers, what are you going to do?
If you have to use specifically even or odd integers, what are you going to do?
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What are consecutive integers?
What are consecutive integers?
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In word problems, x can be the value of
In word problems, x can be the value of
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the difference of 2 rational numbers can be rewritten as
the difference of 2 rational numbers can be rewritten as
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When simplifying non perfect square roots, you keep the square of the
When simplifying non perfect square roots, you keep the square of the
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rational - rational =
rational - rational =
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rational - irrational =
rational - irrational =
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an identity is
an identity is
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The solution to an identity is
The solution to an identity is
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a contradiction is
a contradiction is
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What is a literal equation?
What is a literal equation?
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What is a formula?
What is a formula?
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What happens when solving an inequality and dividing by a negative number?
What happens when solving an inequality and dividing by a negative number?
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When your variable is in the denominator, what should you do?
When your variable is in the denominator, what should you do?
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What happens when you multiply or divide an equation?
What happens when you multiply or divide an equation?
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What is the best option when dealing with literal equations?
What is the best option when dealing with literal equations?
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When is the direction of the inequality symbol reversed?
When is the direction of the inequality symbol reversed?
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What is the solution if variables cancel out and you're left with a true statement in an equation?
What is the solution if variables cancel out and you're left with a true statement in an equation?
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What is the solution if variables cancel out and you're left with a false statement in an equation?
What is the solution if variables cancel out and you're left with a false statement in an equation?
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What is the solution to x < -2 considered?
What is the solution to x < -2 considered?
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Where is it best to have your variable (x) in an equation?
Where is it best to have your variable (x) in an equation?
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Why do all values of x satisfy a true statement in an equality?
Why do all values of x satisfy a true statement in an equality?
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What should you do when your answer needs to make sense in context?
What should you do when your answer needs to make sense in context?
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What should you always do after solving an equation or inequality?
What should you always do after solving an equation or inequality?
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What is a compound inequality?
What is a compound inequality?
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How do you solve a compound inequality?
How do you solve a compound inequality?
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What is the first step of solving a compound inequality?
What is the first step of solving a compound inequality?
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What is the second step of solving a compound inequality?
What is the second step of solving a compound inequality?
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What is the third step of solving a compound inequality?
What is the third step of solving a compound inequality?
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What is the 4th step of solving a compound inequality?
What is the 4th step of solving a compound inequality?
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How do you set notations for a compound inequality?
How do you set notations for a compound inequality?
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When using AND in a compound inequality, what are you looking for on the graph?
When using AND in a compound inequality, what are you looking for on the graph?
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What should you do with the AND/OR when solving a compound inequality?
What should you do with the AND/OR when solving a compound inequality?
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What happens if an AND compound inequality has no intersection on the graph?
What happens if an AND compound inequality has no intersection on the graph?
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What is the solution of compound inequalities with OR?
What is the solution of compound inequalities with OR?
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What is the union in a compound inequality?
What is the union in a compound inequality?
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What happens in an OR compound inequality if arrows point throughout the whole number line?
What happens in an OR compound inequality if arrows point throughout the whole number line?
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What does a compound inequality combine?
What does a compound inequality combine?
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What are the two types of compound inequalities?
What are the two types of compound inequalities?
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If an AND compound inequality makes an OR graph, what does it mean?
If an AND compound inequality makes an OR graph, what does it mean?
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If an OR compound inequality makes an AND graph, what does it mean?
If an OR compound inequality makes an AND graph, what does it mean?
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Where do you always want to isolate x in an AND compound inequality?
Where do you always want to isolate x in an AND compound inequality?
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In a regular AND compound inequality, where should you shade on the graph?
In a regular AND compound inequality, where should you shade on the graph?
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In an AND compound inequality like (4 > x > -2), where should the smaller number come?
In an AND compound inequality like (4 > x > -2), where should the smaller number come?
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How can you ensure the smallest number comes first in an AND compound inequality?
How can you ensure the smallest number comes first in an AND compound inequality?
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What does "<" mean in absolute value equations/inequalities?
What does "<" mean in absolute value equations/inequalities?
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What does "<" mean in absolute value equations/inequalities?
What does "<" mean in absolute value equations/inequalities?
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How can you remember the meaning of < and > in absolute value equations/inequalities?
How can you remember the meaning of < and > in absolute value equations/inequalities?
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What symbols are used to represent absolute value?
What symbols are used to represent absolute value?
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How do you know if you've solved an absolute value inequality correctly?
How do you know if you've solved an absolute value inequality correctly?
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What is the first step in solving absolute value word problems?
What is the first step in solving absolute value word problems?
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What is the second step in solving absolute value word problems?
What is the second step in solving absolute value word problems?
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What is the third step in solving absolute value word problems?
What is the third step in solving absolute value word problems?
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What is the fourth step in solving absolute value word problems?
What is the fourth step in solving absolute value word problems?
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What is the fifth step in solving absolute value word problems?
What is the fifth step in solving absolute value word problems?
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What is the second inequality in an absolute value solution?
What is the second inequality in an absolute value solution?
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What is the sixth step in solving absolute value word problems?
What is the sixth step in solving absolute value word problems?
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What is the seventh step in solving absolute value word problems?
What is the seventh step in solving absolute value word problems?
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What is the eighth step in solving absolute value word problems?
What is the eighth step in solving absolute value word problems?
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What steps do you need to solve regular absolute value inequalities/equations?
What steps do you need to solve regular absolute value inequalities/equations?
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How do you solve an equation of the form |A| = b?
How do you solve an equation of the form |A| = b?
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What does absolute value mean?
What does absolute value mean?
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What are the bars surrounding the expression in an absolute value equation called?
What are the bars surrounding the expression in an absolute value equation called?
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What is the first step when solving absolute value equations or inequalities?
What is the first step when solving absolute value equations or inequalities?
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What do you do after isolating the absolute value expression?
What do you do after isolating the absolute value expression?
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What does the equation |2x - 5| = 13 represent?
What does the equation |2x - 5| = 13 represent?
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How many points are a given distance away from zero on a number line?
How many points are a given distance away from zero on a number line?
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How do you find the value of 'x' in an absolute value equation?
How do you find the value of 'x' in an absolute value equation?
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When do you split an absolute value equation into two equations?
When do you split an absolute value equation into two equations?
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How can you represent both positive and negative values of a number in a solution?
How can you represent both positive and negative values of a number in a solution?
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What is the first equation created when splitting an absolute value equation?
What is the first equation created when splitting an absolute value equation?
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What is the second equation created when splitting an absolute value equation?
What is the second equation created when splitting an absolute value equation?
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Slope of a line
Slope of a line
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Rise
Rise
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Run
Run
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Slope Formula
Slope Formula
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Slope of a horizontal line
Slope of a horizontal line
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Slope-Intercept Form
Slope-Intercept Form
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Slope
Slope
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Y-Intercept
Y-Intercept
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Graphing Slope-Intercept Form
Graphing Slope-Intercept Form
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Finding Slope from Two Points
Finding Slope from Two Points
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Point-Slope Form
Point-Slope Form
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Why is point-slope form useful?
Why is point-slope form useful?
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What is the slope of a line?
What is the slope of a line?
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Find the equation of a line with slope 1/2 passing through (3, -2)
Find the equation of a line with slope 1/2 passing through (3, -2)
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Write the equation in point-slope form given two points (2, -1) and (-3, 3)
Write the equation in point-slope form given two points (2, -1) and (-3, 3)
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What is Standard Form of a linear equation?
What is Standard Form of a linear equation?
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How to find x- and y-intercepts in Standard Form
How to find x- and y-intercepts in Standard Form
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What are horizontal and vertical lines in terms of equations?
What are horizontal and vertical lines in terms of equations?
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How to graph linear equations in Standard Form
How to graph linear equations in Standard Form
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How to translate word problems into linear equations
How to translate word problems into linear equations
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Parallel Lines
Parallel Lines
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Perpendicular Lines
Perpendicular Lines
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Study Notes
Rational and Irrational Numbers
- Rational numbers added/multiplied together result in a rational number.
- Irrational numbers added/multiplied with a rational number results in an irrational number.
- Irrational numbers added/multiplied together may result in a rational or irrational number (it depends).
Real Number System
- The Real Number System contains rational and irrational numbers.
- The symbol R with a line through it represents a Real number.
- A set is a collection of objects, numbers, or elements.
Sets and Subsets
- Sets are represented using braces {}.
- An element of a set is an object within that set.
- A subset only contains elements from the original set.
- The null or empty set is a subset of every set.
Operations with Rationals and Irrationals
- Rational numbers added together are always rational.
- Rational numbers multiplied together are always rational.
- Rational numbers subtracted from each other are always rational.
- A rational number subtracted from an irrational number is always irrational.
- An irrational number added to an irrational number is always irrational.
Special Numbers
- Zero (0) is a rational, integer, and whole number.
- Irrational numbers like pi (Ï€) and imperfect square roots do not have repeating patterns.
Consecutive Integers
- In problems involving consecutive integers, 'x' can represent the first, middle, or last number, depending on the problem's requirements.
- To find consecutive even or odd integers, increment by 2 (x, x+2, x+4...).
Variable Values in Word Problems
- In word problems, 'x' can represent any unknown value.
Simplifying Non-Perfect Square Roots
- When simplifying non-perfect square roots, keep the non-perfect square root factor as a multiple and its square under the radical symbol.
Identities and Contradictions
- An identity is an equation always true for any variable.
- The solution to an identity is all real numbers.
- A contradiction is an equation with no solutions.
Literal Equations
- A literal equation contains more than one variable.
- A formula is a literal equation that shows a relationship between one quantity and one or more other quantities.
Solving Inequalities
- When solving an inequality and dividing or multiplying by a negative number, the inequality sign must be flipped.
Solving Literal Equations
- When a variable is in the denominator and the numerator is another variable in a literal equation, multiply by the variable in the denominator and continue simplifying.
- Try to factor out the variable when possible for a clearer solution.
- When multiplying or dividing both sides of an equation, the operation must be applied to all parts of the equation on both sides.
Solving Equations/Inequalities with Solutions
- When variables cancel out and the result is a true statement (e.g., 5 = 5), the solution is all real numbers.
- When variables cancel out and the result is a false statement (e.g., 5 = 6), there is no solution.
- If the solution is an inequality (like x < -2), there are infinitely many solutions, but those solutions must fit the inequality.
- When solving problems, ideally, the variable will be isolated on the left side.
Checking Answers
- Always check answers when solving equations/inequalities; a solution must make sense in the context given.
- Round results to the nearest whole number, if necessary, to ensure the answer makes sense given the problem.
Compound Inequalities
- A compound inequality combines more than one inequality using the words "and" (or) "or."
- To solve a compound inequality, solve each inequality separately.
- The first step in solving a compound inequality is to solve the first inequality.
- The second step in solving a compound inequality is to solve the second inequality.
- The third step to solving a compound inequality is to combine the two answers into a notation.
- The Fourth step to solving a compound inequality is to graph it on the number line.
- To set the notation, use a brace, place the variable, a wall between the variable and the next portion, write the solution from the first inequality, use "and" or "or", write the solution from the second inequality, and place another brace.
- When using "and" in a compound inequality, look for the intersection on the number line.
- If the compound inequality contains "and" and there is no intersection, there is no solution for the compound inequality.
- When the compound inequality uses "or," find the union of the solution sets.
- The union is the solution where at least one of the inequalities begins to have solutions; the other one will overlap soon after.
- If the compound inequality uses "or" and the solution sets cover the entire number line, all values of the variable are solutions because at least one of the inequalities holds true for all real numbers.
- Compound inequalities combine two or more simple inequalities, typically using "and" or "or."
- If an "and" compound inequality produces an "or" graph, there's likely no solution.
- If an "or" compound inequality produces an "and" graph, the solution is likely all real numbers or the intersection of the inequalities.
- In a standard "and" compound inequality, your variable should be isolated in the middle.
- In an "and" compound inequality, shade the middle (intersection) on the number line.
- In an "and" compound inequality, put the smallest number first.
- If necessary, rearrange the compound inequality, putting the smallest number first.
Absolute Value Equations/Inequalities
- In absolute value equations/inequalities, "<" means AND, and ">" means OR. (LA GO!)
- Absolute value symbols are straight vertical lines (like | |).
- A correctly solved absolute value inequality's graph must work for the original and two other associated inequalities.
- Steps to solve absolute value word problems: 1) define variables, 2) form the absolute value inequality, 3) determine if it's AND or OR, 4) convert to two regular inequalities, 5) Solve for the variable, 6) set the solution notation, 7) graph it, 8) make sure the intersection is shaded for AND (and check your work).
- To solve an absolute value equation of the form |A| = b (where b ≥ 0), solve A = b and A = –b.
- Absolute value represents distance from zero (always positive).
- Absolute value bars are also called modular bars.
- To solve an absolute value equation/inequality, first isolate the absolute value expression.
- Then, create two equations / inequalities. For inequalities, one will be the same as the original but without the absolute value bars, and the other will have the inequality sign flipped. If the value outside the absolute value signs is negative, it has no solution since absolute value cannot be negative.
- If the situation involves distance from a number, add or subtract the distance from the number given to find the possible values.
- A compound inequality is formed for absolute value inequalities. If the inequality is less than (either < or ≤), use the compound inequality type of -b < a < b. If the inequality is greater than (either > or ≥) then the solution is a < -b or a > b.
- If the result of an inequality or equation is something straightforward (such as 0 > -25), the solution is all real numbers since this is a TRUE statement.
- Slope of a line is the ratio of the vertical change (rise) and the horizontal change (run) between any two points on the line.
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