Absolute Value Equalities and Inequalities

• Absolute value: The distance from zero on the number line.

• Example: The absolute value of 7, written as |7|, = 7

• The absolute value of – 3, written as | -3 |, = 3.

• An absolute value is never negative

Solve | x | = 13. Note. There will generally be two answers.

First, solve for the positive• X = 13

Then, negate the right side of the inequality to find the second solution.

• X = - 13

Solve |x – 3| = 6

First, solve for the positive• X – 3 = 6• X = 9

Then, solve for the negative to find the second solution.

• X – 3 = -6• X = -3

Solve 2 + | x | = 20

• Treat the absolute value symbol as a grouping symbol. You must absolutely get the absolute value symbol by itself prior to solving the equation.

• 2 + | x| = 20• | x| = 18• X = 18 or x = -18

Solve |x| - 2 = 2x + 4

• |x| = 2x + 6• X = 2x + 6• X = - 6 • Check in original equation,

not a solution. 4 ≠ -8

Negate opposite side• |x| = 2x + 6• X = - (2x + 6)• X = -2x – 6• 3x = -6• X = -2• Check in original equation.• 0 = 0• This equation has one

solution.

Absolute value inequalities

• Solving absolute value inequalities uses the same principles as solving absolute value equalities. The major issue that must be remembered is to change the direction of the sign when negating the right side of the equation.

Solve |x| > 3

First, solve for the positive• X > 3

Then, multiply right side by -1

• Multiplying by a negative reverses the sign.

• X < -3

• The answer is, x < -3 or x > 3• Note the use of the word

“or”.

Solve |x| < 3

First solve for the positive• X < 3

Then multiply by -1• Remember to reverse the

sign• X > -3• The answer is, x > -3 and x <3• Note the use of the word

“and”.• This may also be written,

-3 < x < 3

Solve |x| + 8 = 6

• First, combine like terms• |x| = -2• No solution

Remember

• The absolute value symbol must absolutely be by itself when solving the equation or inequality

• An absolute value is always greater than or equal to zero.

• For an inequality, you must reverse the sign when multiplying by -1

• Be aware of “and” statements and “or” statements.