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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:
Test regions: