Solve each inequality. Give the solution set using interval notation.
$-8>3x-5>-12$

Verified step by step guidance

1. Recognize that the compound inequality \(-8 > 3x - 5 > -12\) can be split into two separate inequalities: \(-8 > 3x - 5\) and \$3x - 5 > -12$.
2. Solve the first inequality \(-8 > 3x - 5\) by isolating \(x\): add 5 to both sides to get \(-8 + 5 > 3x\), which simplifies to \(-3 > 3x\).
3. Divide both sides of \(-3 > 3x\) by 3 (remember to keep the inequality direction since 3 is positive), resulting in \(-1 > x\) or equivalently \(x < -1\).
4. Solve the second inequality \$3x - 5 > -12\( by adding 5 to both sides: \)3x > -12 + 5\(, which simplifies to \)3x > -7$.
5. Divide both sides of \$3x > -7$ by 3, yielding \(x > -\frac{7}{3}\). Combine this with the first inequality to find the solution set \(-\frac{7}{3} < x < -1\), and express it in interval notation.

Key Concepts

Compound Inequalities

A compound inequality involves two inequalities joined together, often with 'and' or 'or'. In this problem, the inequality -8 > 3x - 5 > -12 means 3x - 5 is simultaneously less than -8 and greater than -12. Solving requires treating it as two separate inequalities and finding the intersection of their solution sets.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable by performing inverse operations, similar to solving equations, but reversing the inequality sign when multiplying or dividing by a negative number. This process helps find the range of values that satisfy the inequality.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate values not included (open interval), while brackets indicate values included (closed interval). It clearly shows the range of values that satisfy the inequality.