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# Graphing Absolute Value Inequalities

Example

Graphing absolute value inequalities

Graph each absolute value inequality.

a) | y - 2x | ≤ 3

b) | x - y | > 1

Solution

a) The inequality | y - 2x | ≤ 3  is equivalent to -3 y - 2x 3, which is equivalent to the compound inequality

y - 2x 3 and y - 2x ≥ -3.

First graph the lines y - 2x = 3 and y - 2x = -3 as shown in figure (a) below. These lines divide the plane into three regions. Test a point from each region in the original inequality, say (-5, 0), (0, 1), and (5, 0):

 | 0 - 2(-5) | ≤ 3 | 1 - 2 Â· 0 | ≤ 3 | 0 - 2 Â· 5 | ≤ 3 10 ≤ 3 1 ≤ 3 10 ≤ 3

Only (0, 1) satisfies the original inequality. So the region satisfying the absolute value inequality is the shaded region containing (0, 1) as shown in figure (b) above. The boundary lines are solid because of the symbol.

b) The inequality | x - y | > 1 is equivalent to

x - y > 1 or x - y < -1.

First graph the lines x - y = 1 and x - y = -1 as shown in figure (a) below. Test a point from each region in the original inequality, say (-4, 0), (0, 0), and (4, 0):

 | -4 - 0 | > 1 | 0 - 0 | > 1 | 4 - 0 | > 1 4 > 1 0 > 1 4 > 1

Because (-4, 0) and (4, 0) satisfy the inequality, we shade those regions as shown in figure (b) above. The boundary lines are dashed because of the > symbol.

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