Solving Absolute Value Equations and Inequalities Lesson 1 – 6 Solving Absolute Value Equations and Inequalities

Objectives Evaluate expressions involving absolute values. Solve absolute value equations and inequalities.

Absolute Value Expressions The absolute value of a number is its distance from 0 on the number line. Since distance is nonnegative, the absolute value of a number is always positive. The symbol |x| is used to represent the absolute value of a number x.

Evaluating Expressions When evaluating expressions that contain absolute values, the absolute value bars act as a grouping symbol. Perform any operations inside the absolute value bars first.

Example 1 Evaluate if y = -3. Fill in -3 for y. Simplify inside the absolute value bars.

Example 2 Solve the following equation. Add 18 to each side. When solving absolute value equations, you MUST check both solutions. In this case, both solutions work.

Calculator Check

Example 3 Solve the following equation. The absolute value must be on one side of the equation by itself. Therefore, we must subtract 9 from each side. The solution to an absolute value can never be negative. Therefore, this equation has no solution. This is the symbol for the empty set.

Example 4 Remember to check both solutions. This time one of them does not work. Example 4 Subtract x from each side. Distribute the -1. Add 3x to each side. Add 2 to each side. Subtract 6 from each side. Divide each side by 2.

Calculator Check

Example 5 Divide each side by 7. Now set up the two equations. Add 13 to each side. Divide each side by 4. Remember to check both solutions.

Calculator Check

Example 6 Divide each side by 3. Check each answer!

Absolute Value Inequality (<) Absolute value inequalities which involve less than ( < ) or less than or equal to ( ≤ ) use the word and. For example, how would we write: Or, we can write Both ways are correct, but I prefer the 2nd one.

Absolute Value Inequality (>) Absolute value inequalities involving greater than ( > ) or greater than or equal to ( ≥ ) use the word or. For example, consider: This is the only option for writing this one besides interval notation.

Example 7 Do we use “and” or “or”? OR 2 6

Example 8 AND -3 -2 1

SOL Problem