How to Solve Integers Inequalities involving Absolute Values?

A step-by-step guide to solving inequalities involving absolute values

An absolute value is the distance between zero and a number. An example of absolute value would be \(|5|\). This shows it’s five spaces from \(0\). Therefore, the absolute value of it is also \(5\).

Solving inequalities involving absolute values can be done in the following steps:

1. Isolate the absolute value expression on one side of the inequality: Start by subtracting or adding the same value on both sides so that the absolute value expression is on its own on one side of the inequality.
2. Split the inequality into two cases: Once the absolute value expression is isolated, split the inequality into two cases, one where the expression inside the absolute value is positive and one where it is negative.
3. Solve for each case: Solve for the positive and negative cases separately. If the expression inside the absolute value is positive, then the inequality remains unchanged. If it’s negative, then reverse the direction of the inequality.
4. Combine the solutions: The solutions from the two cases are combined to form the final solution. This will be the range of values that satisfy the inequality involving absolute values.

Integers inequalities involving absolute values – Example 1

Compare the absolute value of the numbers: \(|-6|,|3|\)
Solution:
Step 1: Simplify the absolute value. \(|-6|=6\) and \(|3|=3\)
Step 2: Now decide which number is greatest, least, or equal.
\(6>3\)

Integers inequalities involving absolute values – Example 2

Compare the absolute value of the numbers: \(|-2|,|2|\)
Solution:
Step 1: Simplify the absolute value. \(|-2|=2\) and \(|2|=2\)
Step 2: Now decide which number is greatest, least, or equal.
\(2=2\)

Exercises for integers inequalities involving absolute values

Compare the absolute value of the numbers.

1. \(\color{blue}{\left|-7\right|,\:\left|-9\right|}\)
2. \(\color{blue}{\left|-15\right|,\:\left|15\right|}\)
3. \(\color{blue}{\left|2+8\right|,\:\left|10\right|\:\:\:}\)

1. \(\color{blue}{\:9>7}\)
2. \(\color{blue}{15=15}\)
3. \(\color{blue}{10=10}\)