1.7 Solve Absolute Value Equations and Inequalities

Absolute value

The absolute value of a number x, written | x|, is the distance the number is from 0 on a number line.

Extraneous solution

An apparent solution that must be rejected because it does not satisfy the original equation.

INTERPRETING ABSOLUTE VALUE EQUATIONS

Equation |x| = |x - 0| = k

Meaning The distance between x and 0 is __k__ .

Graph

Solutions x - 0 = -k or x - 0 = k

x = _-k_ or x = _k_

Equation |x - b| = k

Meaning The distance between x and b is _k_.

Graph

Solutions x - b = -k or x- b = k

x = _b - k_ or x = _b + k_

Example 1

Solve a simple absolute value equation

Solve |x - 3 | = 6. Graph the solution.

|x - 3| = 6 Write original equation.

x - 3 = _-6_ or x - 3 = _6_ Write equivalent equations.

x = _3 - 6_ or x = _3 + 6_ Solve for x.

x = _-3_ or x = _9_ Simplify.

The solutions are _-3_ and _9_ .These are the values of x that are _6_ units away from _3_ on a number line.

Solve the equation. Then graph the solution.

1. |x|= 5 x = -5 or x = 5

2. |x - 5| = 2 x = 3 or x = 7

SOLVING AN ABSOLUTE VALUE EQUATION

Use these steps to solve an absolute value equation | a + b | = c where c > 0.

Step 1 Write two equations: ax + b = _c_ or ax + b = _- c__ .

Step 2 _Solve_ each equation.

Step 3 Check each solution in the original _absolute value_ equation.

Example 2

Solve an absolute value equation

Solve |4x + 10 | = 6x. Check for extraneous solutions.

|4x + 10 | = 6x Write original equation.

4x + 10 = _6x_ or 4x + 10 = _-6x_ Expression can equal _6x_ or _-6x_ .

10 = _2x_ or 10 = _-10x_ Subtract _4x_ from each side.

_5_ = x or _-1_ = x Solve for x.

Check the apparent solutions to see if either is extraneous.

CHECK

|4x + 10 | = 6x |4x + 10 | = 6x

|4(_5_) + 10| ≟ 6(_5_) |4( -1_ ) + 10 | ≟ 6(_-1_)

|_30_| ≟ _30_ |_6_| ≟ _-6_

_30_ = _30_ _6_ ¹ _-6_

The solution is _5_. Reject _-1_ because it is an _extraneous_ solution.

Solve the equation. Check for extraneous solutions.

3. |2x + 5| = 11

x = 3 and x = -8

4. |3x + 18| = 6x

x = 6; x = -2 is an extraneous solution.

ABSOLUTE VALUE INEQUALITIES

In the inequalities below, c > 0.

Inequality Equivalent Form Graph of Solution

|ax + b| _<_ c -c < ax + b < c

|ax + b| _>_ c ax + b < -c or ax + b > c

Example 3

Solve an inequality of the form |ax + b| > c

Solve |2x + 5 | > 3. Then graph the solution.

The absolute value inequality is equivalent to 2x + 5 < _-3_ or 2x + 5 > _3_ .

First Inequality Second Inequality

2x + 5 < _-3_ Write inequalities. 2x + 5 > _3_

2x < _-8_ Subtract _5_ from each side. 2x > _-2_

2x < _-4_ Divide each side by _2_ . x > _-1_

The solutions are all real numbers less than _-4_ or greater than _-1_.

Example 4

Solve an inequality of the form | ax + b | £ c

Solve |x - 1.5| £ 4.5. Then graph the solution.

x - 1.5 £ 4.5 Write inequality.

_-4.5_ £ x - 1.5 £ _4.5_ Write equivalent compound inequality.

_-3.0_ £ x £ _6.0_ Add _1.5_ to each expression.

The solution is between _-3_ and _6_ , inclusive.

Solve the inequality. Then graph the solution.

5. |x - 2| ³ 7 x £ -5 or x ³ 9

6. |4x - 1| < 9 -2 < x < 2.5