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Solving Absolute Value Inequalities

When any of the inequality signs ($> ,\ < ,\ \geq or\ \leq $) is included in an equation in place of equal (=) sign, the equation is referred as inequality equation. An inequality equation with absolute value expression is known as absolute value inequality. Absolute value of x is expressed as $\left |x  \right |$ which ignores sign of x. It means that  $\left |x  \right |$ makes the value assigned in it positive.

Few examples of absolute value inequalities are:

$\left |x-3  \right |\geq 4$,  $\left |2x  \right |< 5$,  $\left |a^{2}+2a -1   \right |\leq a-5$ etc.
Solving an absolute value inequality means to find the values of variable or intervals which satisfies the inequality.

While solving them, we need to follow the following rules:

Case 1: When inequality is $>$ or $\geq $

Absolute Value Inequality

Case 2
: When inequality is $<$ or $\leq $
Absolute Value Inequality Rule

Let us understand the concept of solving absolute inequalities with the help of examples:

Example 1: Solve $\left | x+3 \right |\geq 1$ for x.
Solution: $\left | x+3 \right |\geq 1$
$ x+3\geq 1$ or $ x+3\leq -1$
$ x+3-3\geq 1-3$ or $ x+3-3\leq -1-3$
$ x\geq -2$ or $ x\leq -4$

Example 2: Solve the following inequality
$\left |3x-5  \right |< 10$.
Solution: $\left |3x-5  \right |< 10$
$3x-5< 10$ and $3x-5> -10$
$3x-5+5< 10+5$ and $3x-5+5> -10+5$
$3x< 15$ and $3x> -5$
$x< 5$ and $x> -$$\frac{5}{3}$
$-\frac{5}{3}$$< x< 5$

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