In Exercises 59–94, solve each absolute value inequality. |3 - (2/3)x| > 5

Verified step by step guidance

1. Start by understanding that the inequality involves an absolute value expression: \(|3 - \frac{2}{3}x| > 5\). The absolute value inequality \(|A| > B\) means that either \(A > B\) or \(A < -B\).
2. Set up two separate inequalities based on the definition of absolute value inequalities: 1) \(3 - \frac{2}{3}x > 5\) 2) \(3 - \frac{2}{3}x < -5\)
3. Solve the first inequality \(3 - \frac{2}{3}x > 5\) by isolating \(x\): - Subtract 3 from both sides: \(- \frac{2}{3}x > 2\) - Multiply both sides by the reciprocal of \(-\frac{2}{3}\), which is \(-\frac{3}{2}\). Remember to reverse the inequality sign because you are multiplying by a negative number.
4. Solve the second inequality \(3 - \frac{2}{3}x < -5\) similarly: - Subtract 3 from both sides: \(- \frac{2}{3}x < -8\) - Multiply both sides by \(-\frac{3}{2}\), reversing the inequality sign again.
5. Combine the solutions from both inequalities to express the final solution set for \(x\). This will be the union of the two solution intervals obtained.

Key Concepts

Absolute Value Inequalities

An absolute value inequality involves expressions within absolute value bars and compares them to a number. To solve |A| > B, where B > 0, split it into two inequalities: A > B or A < -B. This approach helps find all values of the variable that satisfy the inequality.

Isolating the Variable

Before solving, isolate the variable expression inside the absolute value. This often involves simplifying the inequality and performing algebraic operations like addition, subtraction, multiplication, or division. Proper isolation ensures accurate splitting into two separate inequalities.

Solving Linear Inequalities

Once the absolute value inequality is split, solve each linear inequality separately. This involves applying standard inequality rules, such as reversing the inequality sign when multiplying or dividing by a negative number. The solution is the union of both inequality solutions.