Absolute Value Inequalities Worksheets

Let us recall that in solving an absolute value equation, we always come up with two or more solutions. Likewise, in solving a linear inequality, we at all times come up with an interval instead of a single value for the variable involved. To solve absolute value inequalities, we are going to merge techniques used for solving absolute value equations and linear inequalities. Absolute value inequalities are those that can be written in any of the following forms: $$\left | x \right | < k$$, $$\left | x \right | > k$$, $$\left | x \right |\leq k$$ or $$\left | x \right |\geq k$$, where $$k$$ is any real number.

Suppose we are confronted with the absolute value inequalities, $$\left|x\right|\geq 2$$. By inspection, the values that can make this inequality true are: $$\left\{2, 3, 4, 5, ... \right\}$$ and $$\left\{...,-5, -4, -3, -2\right\}$$. Algebraically, the solution to $$\left|x\right|\geq 2$$ is the solution to the compound inequalities, $$x\geq 2\;\text{or}\; x\leq -2$$. Similarly, the solution set of $$\left|x\right|\leq 2$$ is $$\left\{-2, -1, 0, 1, 2\right\}$$. This set is the solution to the compound inequality $$-2\leq x\leq 2$$.

From the above examples, a general rule for solving absolute value inequalities can be formulated. To solve an absolute value inequality in a form $$\left | x \right |\geq k$$, express it as “OR compound inequality”, $$x < -k \;\text{or}\; x > k$$. On the other hand, to solve an absolute value inequality in a form, $$\left | x \right |< k$$, express it as “AND compound inequality”, $$-k < x < k$$. These rules also hold true for $$\left | x \right | \geq k$$ and $$\left | x \right |\leq k$$. Then, follow the rules in solving compound inequalities. Lastly, check your answer.

EXAMPLES:

Solve each absolute value inequality. Write the solution set and sketch the graph.

\begin{align*}&1.\;\;\;\left | x-5 \right |<3 \\&2.\;\;\;2\left | x+2 \right |\geq 4\\&3.\;\;\;\left | x+2 \right |\leq 3\\&4.\;\;\;\left | x+2 \right |>3 \end{align*}

ASNWERS: