**Objectives**

- Identify problems that require the use of absolute value.
- Transform absolute value problems into compound inequalities or equations.

**Key Terms**

- Absolute Value – a number’s distance from zero on a number line. ex. |3| = 3 and |-3| = 3 because both 3 and -3 are a total of 3 real spaces away from zero.
- Intersection – the set of numbers that appear in the solution sets of two inequalities.
- Union – the set of all solutions that are found in an or statement.

**Notes**

- Solve: |x| = 2

Answer: x represents a number 2 spaces from zero. It can be negative or positive; so, x = 2 OR x = -2. Both are correct answers.

- Steps to Solve an Absolute Value Equation

Step 1: If necessary, get the absolute value expression by itself on one side of the equation.

Step 2: Rewrite the absolute value equation as two separate equations: one positive and the other negative.

Step 3: Solve each equation separately.

Step 4: After solving, substitute your answers back into the original equation to verify that your solutions are valid.

- ex. |2x + 1| = 5
- Step 1: skip
- Step 2:
- a. 2x + 1 = 5
- b. -(2x + 1) = 5

* VERY IMPORTANT: the one that is negative NEEDS to have the negative sign outside of the parenthesis. You WILL distribute the negative sign to solve it in the next step.

- Step 3:
- a. 2x + 1 = 5 (subtract 1 on both sides)

2x = 4 (then divide by 2 on both sides)

x = 2 - b. -(2x + 1) = 5 (distribute the negative sign into the parenthesis)

-2x – 1 = 5 (add one to both sides)

-2x = 6 (then divide by -2 on both sides)

x = -3

- a. 2x + 1 = 5 (subtract 1 on both sides)
- Step 4: Substitute the answers for x into the original equation and check for validity
- a. |2(2) + 1| = 5

|4 + 1| = 5

|5| = 5 (true) - b. |2(-3) + 1| = 5

|-6 + 1| = 5

|-5| = 5 (true)

- a. |2(2) + 1| = 5

- No Solution
- a. Absolute value is ALWAYS positive, so there is no solution for |x| = -a (“a” is any real number) because absolute values can NEVER be negative.
- ex. 2•|3x – 9| = -6
- Divide both sides by 2 and see if the right side is still a negative: |3x – 9| = -3
- Since the right side is -3 (a negative integer), the absolute value has NO solutions because it can’t equal a negative number.

- ex. 2•|3x – 9| = -6
- b. When you substitute answers back in to check for validity, one may not work out.
- ex. |x – 3| = x + 2
- a. x – 3 = x + 2 (subtract x on both sides)

-3 = 2 (false, impossible: no solution) - b. -(x – 3) = x + 2 (distribute the negative sign)

-x + 3 = x + 2 (add x to both sides)

3 = 2x + 2 (subtract 2 on both sides)

1 = 2x (divide by 2 on both sides)

1/2 = x (true, one solution)

- a. x – 3 = x + 2 (subtract x on both sides)

- ex. |x – 3| = x + 2

- a. Absolute value is ALWAYS positive, so there is no solution for |x| = -a (“a” is any real number) because absolute values can NEVER be negative.
- Absolute Value Inequalities:
**Greater Than**Signs ( > ):**OR**- The solution to the inequality is EITHER / OR, not both (so do NOT use the word: AND).

- Step 1: |x| > 3
- Remember to set this up as a positive and negative (using parenthesis):
- a. x > 3
- b. -(x) > 3 … so, -x > 3 (divide by -1 on both sides and FLIP the sign)

- Step 2: Rewrite the answer where x is positive:
**x < -3 OR x > 3** - Step 3: Graph

- Absolute Value Inequalities:
**Less Than**Signs ( < ):**AND**- The solution to the inequality is between the positive and negative values: AND (you can write “or,” but AND is more appropriate).
- Step 1: |x| > 3
- Remember to set this up as a positive and negative (using parenthesis):
- a. x < 3
- b. -(x) < 3 … so, -x < 3 (divide by -1 on both sides and FLIP the sign)

- Step 2: Rewrite the answer where x is positive: x < 3
**AND**x > -3, but the best way to write this is:**-3 < x < 3**- You will sometimes still see this written with
**OR**: x < 3 or x > -3, but if you graph it, you will see why it is**AND**.

- You will sometimes still see this written with
- Step 3: Graph

- Cheat Sheet: If “a” is any real number:
- |x| < -a is NEVER true (no solution) b/c absolute value is always positive
- |x| > -a is ALWAYS true (all real numbers) b/c absolute value is always positive and positive is greater than negative.
- |x| > +a is SOMETIMES true (some solutions will work, but not others)
- |x| < +a is ALWAYS true (solutions will be between -a and +a values)

- Real World Example
- The ideal length of a particular metal rod is 20.5 cm.
- The measured length may vary from the ideal length by at most 0.045 cm.
- What is the range of acceptable lengths for the rod?
- Setup: | x – 20.5 | ≤ 0.045 (use a negative sign in the setup b/c you are looking for the
*difference*!- Create both inequalities (positive and negative):
- a. (x – 20.5) ≤ 0.045
- b. – (x – 20.5) ≤ 0.045 which is simplified to: -x + 20.5 ≤ 0.045

- Solve both inequalities with an
**AND**result - So, the range is between the two values: 20.455 ≤ x ≤ 20.545

- Create both inequalities (positive and negative):