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Solving Inequalities Containing Absolute Value

To solve an inequality containing an absolute value, treat the "<",
"≤", ">", or "≥" sign as an "=" sign, and solve the equation
as in Absolute Value Equations. The resulting values of *x*
are called boundary points or critical points.

Plot the boundary points on the number line, using closed circles if
the original inequality contained a ≤ or ≥ sign, and open
circles if the original inequality contained a < or > sign. If you
are unsure which type of circle to use, test each critical point in
the original inequality; if it satisfies the inequality, use a closed
circle.

If there are 2 boundary points, the number line will be divided into 3
regions. Pick a point in each region--not a critical point--and
test this value in the original inequality. If it satisfies the
inequality, draw a dark line over the entire region; if one
point in a region satisfies the inequality, all the points in that
region will satisfy the inequality. Make sure that each region is
tested, because the solution set may consist of multiple regions.

*Example 1*: Solve and graph: | *x* + 1| < 3.

Solve | *x* + 1| = 3:

- Inverse operations: None to reverse.
- Separate:
*x* + 1 = 3 or *x* + 1 = - 3.
- Solve:
*x* = 2 or *x* = - 4.
- Check: | 2 + 1| = 3 ? Yes. | - 4 + 1| = 3 ? Yes.

Thus, the critical points are

*x* = 2 and

*x* = - 4. Graph these as

*open* circles:

Critical Points of | *x* + 1| < 3

Test regions:

Left: *x* = - 5: | - 5 + 1| < 3 ? No.

Middle: *x* = 0: | 0 + 1| < 3 ? Yes.

Right: *x* = 3: | 3 + 1| < 3 ? No.

Graph the inequality:

Graph of | *x* + 1| < 3

*Example 2*: Solve and graph: 4| 2*x* - 1|≥20.

Solve 4| 2*x* - 1| = 20:

- Inverse operations: | 2
*x* - 1| = 5.
- Separate: 2
*x* - 1 = 5 or 2*x* - 1 = - 5.
- Solve:
*x* = 3 or *x* = - 2.
- Check: 4| 2(3) - 1| = 20 ? Yes. 4| 2(- 2) - 1| = 20 ? Yes.

Thus, the critical points are

*x* = 3 and

*x* = - 2. Graph these as

*closed* circles:

Critical Points of 4| 2*x* - 1|≥20

Test regions:

Left: *x* = - 3: 4| 2(- 3) - 1|≥20 ? Yes.

Middle: *x* = 0: 4| 2(0) - 1|≥20 ? No.

Right: *x* = 4: 4| 2(4) - 1|≥20 ? Yes.

Graph the inequality:

Graph of 4| 2*x* - 1|≥20