Solving for Absolute Value Inequality Solution Sets

     You will encounter absolute value inequality problems in various math courses, such as Algebra I, Algebra II and College Algebra.  If you happen to find them a bit confusing, hopefully, the explanation below will help clear things up.

          Absolute Value Inequalities Containing < (“AND” Inequalities)

     Suppose we need to solve for the solution of the following equation:

    \[ |x|\leq 4 \]

     The problem is asking what values of x have an absolute value less or equal to 4. The solution set, therefore, is the set of all real numbers within 4 units of zero on the number line. In other words, all the numbers greater than or equal to -4 (the Negative Endpoint) and all the numbers less than or equal to +4 (the Positive Endpoint).

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     We write the solution set as

We write the solution set as:

    \[\{x | -4 \leq x \leq 4\}  \]

or

    \[\{x | -4 \leq x AND x \leq x \}\]

     Let’s look at a more complex “AND” Inequality:

    \[ 2|x-4|-8 \leq 4 \]

     Just as you do when solving absolute value equations, we want to first isolate the absolute value expression.

    \begin{equation*} 2|x-4| - 8 \leq 4 \end{equation*}

    \begin{equation*} 2|x-4| - 8 +\textbf{8} \leq 4 +\textbf{8} \end{equation*}

    \begin{equation*}2|x+4| \leq 12 \end{equation*}

    \begin{equation*}\frac{2|x+4}{\textbf{2}} \leq \frac{12}{\textbf{2}} \end{equation*}

    \begin{equation*} |x-4| \leq 6 \end{equation*}

    As with the earlier problem, we have to analyze the Negative Endpoint and the Positive Endpoint. The Negative Endpoint shows where the absolute value expression begins in the negative direction and the Positive Endpoint shows where the absolute value expression begins in the positive direction.

Negative Endpoint

    \[x-4 \geq \textbf{-6}\]

    \[x-4 +\textbf{4} \geq -6 + \textbf{4} \]

    \[x \geq -2\]

Postive Endpoint

    \[x-4 \leq \textbf{6} \]

    \[x-4 + \textbf{4} \leq 6 + \textbf{4} \]

    \[x \leq 10\]

     So, x ≥ -2 AND x ≤ 10.  The solution set is {x| -2 ≤ x ≤10} This solution set plots as follows on the number line:

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             Absolute Value Inequalities Containing > (“OR” Inequalities)

     Let’s look at the absolute value inequality problem | x | ≥ 4.

     The problem is asking what values of x have an absolute value greater than4. So, we are looking for all real numbers whose distance from zero is greater than or equal to 4. That means we want all the numbers less than or equal to -4 (the Negative Endpoint) OR all the numbers greater than or equal to +4 (the Positive Endpoint). The solution set is: {x| x ≤ -4 OR x ≥ 4}

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     Let’s look at a more complex “OR” Inequality:

             2|x + 4| – 2 ≥ 10 

     As before, we want to first isolate the absolute value expression.

             2|x + 4| – 2 + 2 ≥ 10 + 2

             2|x + 4| ≥ 12

              22|x + 4| ≥ 122

                |x + 4| ≥ 6

     Now, we analyze the Negative Endpoint and the Positive Endpoint. Remember, the Negative Endpoint is the endpoint for the solution set numbers less than or equal to -6, and the Positive Endpoint is the endpoint for the solution set numbers greater than or equal to +6.

        Negative Endpoint                                                       Positive Endpoint

              x + 4 ≤ -6                                                                         x + 4 ≥ 6   x + 4 – 4 ≤ -6 – 4                         x + 4 – 4 ≥ 6 – 4

               x ≤ -10                                         OR                                     x ≥ 2                              

The solution set is {x| x ≤ -10 or x ≥ 2}

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     I hope this explanation helps you better understand absolute value inequalities.  My next post will address Absolute Value Inequalities with Special Case Solution Sets.

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