12 Questions
What is the first step in solving an inequality?
Isolate the variable(s) by performing inverse operations on both sides of the inequality.
What is the purpose of checking real solutions when solving an inequality?
To determine if the result is true by plugging the isolated variable back into the original inequality or equation.
How can you represent the solution to an inequality?
Use brackets or parentheses to indicate inclusion or exclusion of the endpoint.
What is the purpose of isolating variables in inequalities?
Isolating variables in inequalities is important for solving equations.
What is the second step in solving an equation involving an inequality?
Use the properties of inequalities to simplify and eliminate any unnecessary terms.
How can you solve the inequality $x^2 + 4 > x$?
First, subtract 4 from both sides to get $x^2 > x$. Then, factor out $x$ from the left side to get $x(x - 1) > 0$. Since $x(x - 1)$ can only be positive when $x < 1$ or $x > 1$, the solution is $x < 1$ or $x > 1$.
What happens to the direction of an inequality symbol when adding or subtracting the same value from both sides?
It does not change
How does multiplying or dividing each side of an inequality by a positive number affect the directionality of the inequality?
It maintains the directionality
What happens to the direction of an inequality symbol when dividing each side of an inequality by a negative number?
It reverses
What property states that if 'x < y' and 'y < z', then 'x < z'?
Transitive property
Explain the transitive property with respect to inequalities.
If x is less than y and y is less than z, then x is less than z.
What concept is useful for expressing the solution of an inequality in interval notation?
Interval notation
Study Notes
Inequalities: Math Operations, Number Properties, Isolating Variables, and Solving Equations
An inequality is a relationship between two values or expressions. It compares the magnitudes of two things without specifying exact equality. Some common inequality symbols include "less than" (<), "greater than" (>), "less than or equal to" (≤), and "greater than or equal to" (≥). Rules for operating on inequalities include addition and subtraction by the same quantity maintaining directionality; multiplication and division by positive numbers maintain directionality; division by negative numbers flips directionality.
Math Operations
When working with inequalities, some basic math operations include adding, subtracting, multiplying, and dividing. Adding or subtracting the same value from both sides of an inequality does not change the direction of the inequality symbol. Multiplying or dividing each side of an inequality by a positive number maintains the directionality of the inequality. However, dividing each side of an inequality by a negative number reverses the directionality of the inequality.
Number Properties
Understanding properties of numbers is crucial for solving mathematical problems involving inequalities. For example, the transitive property states that if 'x < y' and 'y < z', then 'x < z'. This means that if x is less than y and y is less than z, then x is less than z. Interval notation is another useful concept for expressing the solution of an inequality in interval notation. To represent the solution, find the values that make the inequality true and use brackets or parentheses to indicate inclusion or exclusion of the endpoint.
Isolating Variables
Isolating variables in inequalities is important for solving equations. To isolate a variable, perform inverse operations on both sides of the equation. For example, to solve an inequality like '2x + 3 > 6', subtract 2 from both sides to get '2x > 3'. Similarly, in an inequality like '5 - 2x = 13', subtract 13 from both sides to get '-2x = 8', then divide both sides by -2 to get '2x = -4'. Remember to check for real solutions by plugging the isolated variable back into the original inequality or equation and checking if the result is true.
Solving Equations
To solve equations involving inequalities, follow these steps:
- Isolate the variable(s) by performing inverse operations on both sides of the inequality.
- Use the properties of inequalities to simplify and eliminate any unnecessary terms.
- Check for real solutions by plugging the isolated variable back into the original inequality or equation and determining if the result is true. If not, adjust your inequality accordingly.
For example, to solve 'x^2 + 4 > x', first subtract 4 from both sides to get 'x^2 > x'. Then, factor out 'x' from the left side to get 'x(x - 1) > 0'. Since 'x(x - 1)' can only be positive when x < 1 or x > 1, the solution is x < 1 or x > 1.
Test your knowledge of inequalities, math operations, number properties, isolating variables, and solving equations. Learn about the rules for operating on inequalities, properties of numbers, isolating techniques, and steps to solve equations involving inequalities.
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