Solving Absolute Value Equations And Inequalities - Math Planet

Algebra 1 / Linear inequalities / Solving absolute value equations and inequalities Do excercises Show all 3 exercises

• Absolute equations I
• Absolute equations II
• Absolute equations III

The absolute number of a number a is written as

$$\left | a \right |$$

And represents the distance between a and 0 on a number line.

An absolute value equation is an equation that contains an absolute value expression. The equation

$$\left | x \right |=a$$

Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

To solve an absolute value equation as

$$\left | x+7 \right |=14$$

You begin by making it into two separate equations and then solving them separately.

$$x+7 =14$$

$$x+7\, {\color{green} {-\, 7}}\, =14\, {\color{green} {-\, 7}}$$

$$x=7$$

or

$$x+7 =-14$$

$$x+7\, {\color{green} {-\, 7}}\, =-14\, {\color{green} {-\, 7}}$$

$$x=-21$$

An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

The inequality

$$\left | x \right |<2$$

Represents the distance between x and 0 that is less than 2

Whereas the inequality

$$\left | x \right |>2$$

Represents the distance between x and 0 that is greater than 2

You can write an absolute value inequality as a compound inequality.

$$\left | x \right |<2\: or

$$-2<x<2$$

This holds true for all absolute value inequalities.

$$\left | ax+b \right |<c,\: where\: c>0$$

$$=-c<ax+b<c$$

$$\left | ax+b \right |>c,\: where\: c>0$$

$$=ax+b<-c\: or\: ax+b>c$$

You can replace > above with ≥ and < with ≤.

When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality.

Example

Solve the absolute value inequality

$$2\left |3x+9 \right |<36$$

$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$

$$\left | 3x+9 \right |<18$$

$$-18<3x+9<18$$

$$-18\, {\color{green} {-\, 9}}<3x+9\, {\color{green} {-\, 9}}<18\, {\color{green} {-\, 9}}$$

$$-27<3x<9$$

$$\frac{-27}{{\color{green} 3}}<\frac{3x}{{\color{green} 3}}<\frac{9}{{\color{green} 3}}$$

$$-9<x<3$$

Video lesson

Solve the absolute value equation

$$4 \left |2x -1 \right | -2 = 10$$

Do excercises Show all 3 exercises

• Absolute equations I
• Absolute equations II
• Absolute equations III

More classes on this subject Algebra 1 Linear inequalities: Solving linear inequalities Algebra 1 Linear inequalities: Solving compound inequalities

Next Chapter:

LINEAR INEQUALITIES – Linear inequalities in two variables Search

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