Algebra 1 Mid-Term Review Pt. 1 (Topics 1 & 2)
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Questions and Answers

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?

rational

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?

<p>irrational</p> Signup and view all the answers

What is the name for the specific number system that encompasses both rational and irrational numbers?

<p>The Real Number System</p> Signup and view all the answers

What does the symbol 'R' with a line through it represent?

<p>Real number</p> Signup and view all the answers

What is a set, in mathematical terms?

<p>A collection of objects such as numbers</p> Signup and view all the answers

What is an element of a set?

<p>An object within that set</p> Signup and view all the answers

What symbols are used to list sets?

<p>Braces, {}</p> Signup and view all the answers

What is a subset of a set?

<p>It only contains elements from that set</p> Signup and view all the answers

What type of number is the sum of two rational numbers?

<p>rational</p> Signup and view all the answers

What type of number is the product of two rational numbers?

<p>rational</p> Signup and view all the answers

What type of number is the sum of a rational number and an irrational number?

<p>irrational</p> Signup and view all the answers

What type of number is the product of a rational number and an irrational number?

<p>irrational</p> Signup and view all the answers

Which of the following number sets does 0 belong to?

<p>All of the above</p> Signup and view all the answers

What is a defining characteristic of an irrational number?

<p>It has no patterns and you don't know how it's gonna end</p> Signup and view all the answers

What are some examples of irrational numbers?

<p>pi and imperfect square roots</p> Signup and view all the answers

In consecutive integer problems, what can 'x' represent?

<p>The first, middle, or last number (depends on what you want it to be)</p> Signup and view all the answers

If you need to use specifically even or odd integers, what adjustments do you make to 'x' when representing consecutive integers?

<p>You're going to add 2 to x each time. For example, it would be: x + x+2 + x+4</p> Signup and view all the answers

What are consecutive integers?

<p>Numbers that follow each other on the number line</p> Signup and view all the answers

In word problems, 'x' can represent the value of what?

<p>Whatever you want it to be</p> Signup and view all the answers

How can the difference of two rational numbers be rewritten?

<p>A single ratio</p> Signup and view all the answers

When simplifying non-perfect square roots, what factor should you keep?

<p>The non-perfect square root factor</p> Signup and view all the answers

What type of number is the difference of two rational numbers?

<p>rational</p> Signup and view all the answers

What type of number is the difference of a rational number and an irrational number?

<p>irrational</p> Signup and view all the answers

What is an identity?

<p>An equation that is true for any variable; meaning no matter what value is plugged in, the equation will be true. These equations are equal to themselves.</p> Signup and view all the answers

What is the solution set for an identity?

<p>All real numbers!</p> Signup and view all the answers

What is a contradiction?

<p>An equation that has no solution; there are no values of a variable that make it true. These equations are equal to 0.</p> Signup and view all the answers

What does a literal equation contain?

<p>More than one variable</p> Signup and view all the answers

Describe what a formula is.

<p>An equation that expresses a relationship between one quantity and one or more other quantities</p> Signup and view all the answers

What happens when you divide an inequality by a negative number?

<p>You must flip the inequality sign.</p> Signup and view all the answers

What is the recommended approach to solving a literal equation when your variable is in the denominator and the numerator has another variable?

<p>Multiply both sides of the equation by the variable in the denominator.</p> Signup and view all the answers

When you multiply or divide an equation by a value, what needs to be done to ensure accuracy?

<p>Multiply or divide all parts of the equation, regardless of whether they are on the same or opposite sides.</p> Signup and view all the answers

What is the best strategy to use with literal equations when possible?

<p>Factor out the variable you are solving for.</p> Signup and view all the answers

Under what circumstance does the direction of the inequality symbol get reversed in an inequality?

<p>When dividing or multiplying by a negative value.</p> Signup and view all the answers

What does the solution being all real numbers imply in an equation, when variables cancel out and you are left with a true statement?

<p>Any real number can substitute for the variable, making the equation true.</p> Signup and view all the answers

What does it indicate when variables cancel out in an equation, resulting in a false statement?

<p>There is no solution to the equation.</p> Signup and view all the answers

Explain what kind of solutions does the inequality x < -2 have?

<p>Infinitely many solutions that are less than -2.</p> Signup and view all the answers

Where is it ideal to have the variable x (or any variable) positioned in an equation?

<p>On the left side.</p> Signup and view all the answers

Why are all values of x true in an equality?

<p>Because regardless of what value you plug in for x, the resulting values will always be consistent with the relationship established by the inequality.</p> Signup and view all the answers

What is the general rule for rounding answers in word problems, keeping in mind the context of the problem?

<p>Round to the nearest whole number that makes sense within the context of the problem.</p> Signup and view all the answers

What is a crucial step after solving a mathematical problem?

<p>Always check your answer.</p> Signup and view all the answers

What is a compound inequality?

<p>An inequality that combines more than one inequality using the words 'and' or 'or'.</p> Signup and view all the answers

What is the first step in solving a compound inequality?

<p>Solve each individual inequality separately.</p> Signup and view all the answers

What is the second step in solving a compound inequality?

<p>Solve the second inequality.</p> Signup and view all the answers

Describe the process of setting up a notation for the solution of a compound inequality.

<p>Put a brace, include the variable, put a wall between the variable and the next part of the notation, include the solution to the first inequality, include either 'and' or 'or', include the solution to the second inequality, and finally, put another brace.</p> Signup and view all the answers

In a compound inequality with AND, what does the intersection represent on the graph?

<p>The intersection of the shaded regions, representing values that satisfy both inequalities.</p> Signup and view all the answers

How should you handle the 'and' or 'or' in a compound inequality throughout the solving process?

<p>Bring them down consistently throughout the steps of solving the inequality.</p> Signup and view all the answers

If there is no intersection in the graph of an AND compound inequality, what does that indicate about the solution?

<p>It indicates that there is no solution to the compound inequality.</p> Signup and view all the answers

What does the solution of OR compound inequalities represent on the graph?

<p>The union of the shaded regions, representing values that satisfy at least one of the inequalities.</p> Signup and view all the answers

Describe the union as it pertains to compound inequalities.

<p>The range of values where at least one of the inequalities begins to have solutions and the other inequality's solutions come in soon after.</p> Signup and view all the answers

If a compound inequality with OR has arrows pointing throughout the entire number line, what does this mean about the solution set?

<p>All values of the variable are solutions because all real numbers satisfy at least one of the inequalities.</p> Signup and view all the answers

What does a compound inequality combine?

<p>Two simple inequalities.</p> Signup and view all the answers

What are the two types of compound inequalities?

<p>AND &amp; OR</p> Signup and view all the answers

If an AND compound inequality results in an OR graph, what is the implication?

<p>This rarely happens, and it means there is no solution.</p> Signup and view all the answers

If an OR compound inequality results in an AND graph, what does that mean?

<p>This rarely happens, and the answer is ALL REAL NUMBERS.</p> Signup and view all the answers

In which type of compound inequality should you always isolate x (or the variable) in the middle?

<p>AND compound inequalities.</p> Signup and view all the answers

When graphing a regular AND compound inequality, which areas should be shaded?

<p>Only the middle region, or the intersection of the shaded areas.</p> Signup and view all the answers

In an AND compound inequality like 4 > x > -2, how do you approach the number arrangement?

<p>You want the smallest number to come first.</p> Signup and view all the answers

What can you do to ensure the smallest number comes first in an AND compound inequality?

<p>Rearrange the compound inequality to get the smaller number first.</p> Signup and view all the answers

In Absolute Value Equations/Inequalities, which symbol means AND?

<p>less than (regarding the variable)</p> Signup and view all the answers

How would you remember that Less than (regarding the variable) means AND and Greater than (regarding the variable) means OR?

<p>LA GO!</p> Signup and view all the answers

What are the absolute value symbols?

<p>Walls that look like (l) but completely straight</p> Signup and view all the answers

You know you solved an absolute value inequality correctly if the graph works for the original and two other inequalities.

<p>True</p> Signup and view all the answers

What is the first step to solve absolute value inequalities/equations word problems?

<p>determine the variables (what they are and how you're going to represent them)</p> Signup and view all the answers

What is the second step to solve absolute value inequalities/equations word problems?

<p>Come up with your absolute value inequality using your variable and the absolute value symbols</p> Signup and view all the answers

Which steps are necessary to solve a regular absolute value inequality/equation without a word problem?

<p>#3 - #8</p> Signup and view all the answers

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.

<p>True</p> Signup and view all the answers

What does absolute value mean?

<p>the distance of a number from zero (this is always positive)</p> Signup and view all the answers

What are the bars indicating absolute value called?

<p>absolute value or modular bars</p> Signup and view all the answers

What is the first step to solving any equation/inequality that involves absolute value?

<p>isolate the absolute value expression</p> Signup and view all the answers

What do you do after isolating the absolute value expression in an equation or inequality?

<p>create TWO EQUATIONS</p> Signup and view all the answers

What does the equation the absolute value of 2x - 5 = 13 mean?

<p>the distance on a number line from 2x - 5 to 0 is 13 units</p> Signup and view all the answers

How many points are 13 units away from zero? What are they?

<p>2, 13, -13</p> Signup and view all the answers

What do you do to solve for x in the equation the absolute value of 2x - 5 = 13?

<p>solve the equations</p> Signup and view all the answers

When do you split the absolute value equation up into two separate ones?

<p>when the absolute value equation is completely isolated on one side of the equation</p> Signup and view all the answers

How would you represent that both positive and negative values of a number work for the variable?

<p>-+(whatever #) (ex. -+9)</p> Signup and view all the answers

In the broken up equations, what is the first equation?

<p>the same as the absolute value equation, with no absolute value symbols</p> Signup and view all the answers

In the broken up equations, what is the second equation?

<p>the opposite of the first (previous) expression</p> Signup and view all the answers

Any equation in which the absolute value expression equals a negative value has no solution.

<p>True</p> Signup and view all the answers

Why does an absolute value equation have no solution when it equals a negative value?

<p>absolute value cannot be negative</p> Signup and view all the answers

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.

<p>True</p> Signup and view all the answers

What do Absolute Value Inequalities turn into?

<p>compound inequalities</p> Signup and view all the answers

How should you write the solution of an AND absolute value inequality?

<p>(ex. -1 &lt; n &lt; 3)</p> Signup and view all the answers

Once you isolate the absolute value inequality, what is the next step?

<p>you can THEN determine if it's AND or OR</p> Signup and view all the answers

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?

<p>solve the compound inequality -b &lt; (or equal to) A &lt; (or equal to) b</p> Signup and view all the answers

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?

<p>solve the compound inequality A &lt; -b OR A &gt; b</p> Signup and view all the answers

Irrational/Rational = Irrational

<p>True</p> Signup and view all the answers

Could 4.137604827 be rational?

<p>True</p> Signup and view all the answers

What do you have to do to solve consecutive integer problems?

<p>do x + x + 1 + x + 2</p> Signup and view all the answers

What is the solution to 0 > -25 considered?

<p>ALL REAL NUMBERS because it is a TRUE STATEMENT</p> Signup and view all the answers

What is a good thing to do when solving absolute value equations/inequalities?

<p>GRAPH WHENEVER YOU CAN</p> Signup and view all the answers

What is a good thing to keep in mind when solving absolute value equations/inequalities?

<p>KEEP IN MIND WHETHER IT'S AN AND OR AN OR</p> Signup and view all the answers

What is the most accurate result?

<p>the exact one (even if it's an imperfect square)</p> Signup and view all the answers

What would be the most appropriate solution?

<p>one that clearly and reasonably represent the solution</p> Signup and view all the answers

Be on the look out for what scenario?

<p>being left with an inequality or equation without variables</p> Signup and view all the answers

What should you do if you get something about x's distance from another number being a certain amount of units (absolute value).

<p>Add and subtract the amount of units from the number, giving you what the numbers will be, then you just have to make the right equation. This would be something like absolute value of x - 4 = 13.</p> Signup and view all the answers

What is the formula to find the slope between two points (x1, y1) and (x2, y2)?

<p>m = (y2 - y1) / (x2 - x1)</p> Signup and view all the answers

A slope of 0 indicates a vertical line.

<p>False</p> Signup and view all the answers

What does 'rise' refer to when calculating slope?

<p>The difference in y-values of two points.</p> Signup and view all the answers

The formula for calculating slope is m = (y2 - y1) / (x2 - ______).

<p>x1</p> Signup and view all the answers

Match the following types of slope with their description:

<p>Positive Slope = Line rises from left to right Negative Slope = Line falls from left to right Zero Slope = Horizontal line Undefined Slope = Vertical line</p> Signup and view all the answers

In the equation $y = mx + b$, what does the variable 'm' represent?

<p>The slope of the line</p> Signup and view all the answers

The y-intercept of a graph is the point at which the line crosses the x-axis.

<p>False</p> Signup and view all the answers

What is the slope-intercept form of the equation for a line that passes through the point (0, 1) and (2, 2)?

<p>y = 0.5x + 1</p> Signup and view all the answers

In the function $y = 3x + 4$, the y-intercept is ______.

<p>4</p> Signup and view all the answers

Match the following equations with their corresponding slopes:

<p>y = 2x + 1 = Slope = 2 y = -3x + 5 = Slope = -3 y = 0.5x - 4 = Slope = 0.5 y = x + 3 = Slope = 1</p> Signup and view all the answers

What is the point-slope form equation of a line with a slope of $\frac{1}{2}$ that passes through the point (3, -2)?

<p>y + 2 = $\frac{1}{2}$(x - 3)</p> Signup and view all the answers

The equation y - 3 = 1(x + 1) represents a line with a slope of 1.

<p>True</p> Signup and view all the answers

Write an equation in point-slope form for the line that goes through the points (2, -1) and (-3, 3).

<p>y + 1 = -\frac{4}{5}(x - 2)</p> Signup and view all the answers

In point-slope form, the equation is expressed as y - ______ = m(x - ______).

Signup and view all the answers

What is the point-slope form of a linear equation?

<p>y - y1 = m(x - x1)</p> Signup and view all the answers

The slope-intercept form is derived from the point-slope form.

<p>True</p> Signup and view all the answers

Write the equation of a line in point-slope form that has a slope of 1/2 and passes through the point (3, -2).

<p>y + 2 = 1/2(x - 3)</p> Signup and view all the answers

The graph of the equation y - 3 = 1(x + 1) will cross the y-axis at ______.

<p>4</p> Signup and view all the answers

Match the following point-slope equations with their corresponding slopes:

<p>y - 2 = 3(x - 1) = 3 y + 1 = -2(x - 2) = -2 y - 4 = 1/2(x + 3) = 1/2 y + 3 = 0(x - 5) = 0</p> Signup and view all the answers

What is the standard form of a linear equation?

<p>Ax + By = C</p> Signup and view all the answers

The y-intercept of a line is where it intersects the x-axis.

<p>False</p> Signup and view all the answers

In the equation 3x + 4y = 12, what are the intercepts if x=0?

<p>4</p> Signup and view all the answers

The x-intercept occurs when y equals ______.

<p>0</p> Signup and view all the answers

Match the following variables with their descriptions:

<p>x = Number of questions worth 2 points y = Number of questions worth 4 points A = Coefficient of x in standard form B = Coefficient of y in standard form</p> Signup and view all the answers

Which of the following slopes represent parallel lines?

<p>3</p> Signup and view all the answers

All vertical lines are considered parallel.

<p>True</p> Signup and view all the answers

What is the slope of a line that is perpendicular to a line with a slope of 2?

<p>-1/2</p> Signup and view all the answers

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 + ___

<p>3</p> Signup and view all the answers

Match the following pairs of lines to their classification:

<p>4x + 3y = 6 = y = -4/3x + 2 y = -5x = 25x + 5y = 1 y = 2x = y = 2x + 1 y = 3x + 1 = y = 3x - 4</p> Signup and view all the answers

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 (π) 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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