How to Put an Absolute Value Equation or Inequality on a Number Line

Absolute value equations have two distinct solutions on the number line.
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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.

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