Solve each inequality. Give the solution set in interval notation. -4≤(x+1)/2≤5

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Start by understanding that the compound inequality \(-4 \leq \frac{x+1}{2} \leq 5\) means that \(\frac{x+1}{2}\) is between \(-4\) and \(5\), inclusive.

To isolate \(x\), first eliminate the denominator by multiplying all parts of the inequality by 2: \(2 \times (-4) \leq 2 \times \frac{x+1}{2} \leq 2 \times 5\), which simplifies to \(-8 \leq x+1 \leq 10\).

Next, subtract 1 from all parts of the inequality to isolate \(x\): \(-8 - 1 \leq x + 1 - 1 \leq 10 - 1\), which simplifies to \(-9 \leq x \leq 9\).

Interpret the solution: \(x\) is greater than or equal to \(-9\) and less than or equal to \(9\).

Express the solution set in interval notation as \([-9, 9]\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Compound Inequalities

Compound inequalities involve two inequalities combined into one statement, often using 'and' or 'or'. In this problem, the inequality -4 ≤ (x+1)/2 ≤ 5 means both conditions must be true simultaneously, so the solution set includes values of x that satisfy both inequalities at once.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable by performing inverse operations, similar to solving equations, but with attention to inequality direction. Multiplying or dividing by a positive number keeps the inequality direction the same, while doing so by a negative number reverses it.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Brackets [ ] indicate inclusion of endpoints, while parentheses ( ) indicate exclusion. For example, [a, b] means all values from a to b, including a and b.