How to graph absolute value inequalities on a number line

Absolute value equations and inequalities add a twist to algebraic solutions, allowing the solution to be either the positive or negative value of a number. Graphing absolute value equations and inequalities is a more complex procedure than graphing regular equations because you have to simultaneously show the positive and negative solutions. Simplify the process by splitting the equation or inequality into two separate solutions before graphing.

Absolute Value Equation

Isolate the absolute value term in the equation by subtracting any constants and dividing any coefficients on the same side of the equation. For example, to isolate the absolute variable term in the equation 3|x - 5| + 4 = 10, you would subtract 4 from both sides of the equation to get 3|x - 5| = 6, then divide both sides of the equation by 3 to get |x - 5| = 2.

Split the equation into two separate equations: the first with the absolute value term removed, and the second with the absolute value term removed and multiplied by -1. In the example, the two equations would be x - 5 = 2 and -(x - 5) = 2.

Isolate the variable in both equations to find the two solutions of the absolute value equation. The two solutions to the example equation are x = 7 (x - 5 + 5 = 2 + 5, so x = 7) and x = 3 (-x + 5 - 5 = 2 - 5, so x = 3).

Draw a number line with 0 and the two points clearly labeled (make sure the points increase in value from left to right). In the example, label points -3, 0 and 7 on the number line from left to right. Place a solid dot on the two points corresponding to the solutions of the equation found in Step 3 -- 3 and 7.

Absolute Value Inequality

Isolate the absolute value term in the inequality by subtracting any constants and dividing any coefficients on the same side of the equation. For example, in the inequality |x + 3| / 2 < 2, you would multiply both sides by 2 to remove the denominator on the left. So |x + 3| < 4.

Split the equation into two separate equations: the first with the absolute value term removed, and the second with the absolute value term removed and multiplied by -1. In the example, the two inequalities would be x + 3 < 4 and -(x + 3) < 4.

Isolate the variable in both inequalities to find the two solutions of the absolute value inequality. The two solutions to the previous example are x < 1 and x > -7. (You must reverse the inequality symbol when multiplying both sides of an inequality by a negative value: -x - 3 < 4; -x < 7, x > -7.)

Draw a number line with 0 and the two points clearly labeled. (Make sure the points increase in value from left to right.) In the example, label points -1, 0 and 7 on the number line from left to right. Place an open dot on the two points corresponding to the solutions of the equation found in Step 3 if it is a < or > inequality and a filled dot if it is a ≤ or ≥ inequality.

Draw solid lines visibly thicker than the number line to show the set of values that the variable can take. If it is a > or ≥ inequality, make one line extend to negative infinity from the lesser of the two dots and another line extending to positive infinity from the greater of the two dots. If it is a < or ≤ inequality, draw a single line connecting the two dots.

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

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

How do you graph inequalities on a number line?