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Chapter 3: Problem 49

Solve each inequality. $$-5 \leq-4 y+3<7$$

Short Answer

Expert verified
The solution is \(-1 < y \leq 2\).

Step by step solution

01

Break Down the Compound Inequality

Consider the compound inequality \(-5 \leq -4y + 3 < 7\). It represents two separate inequalities:1. \(-5 \leq -4y + 3\)2. \(-4y + 3 < 7\).
02

Solve the First Inequality

Solve \(-5 \leq -4y + 3\):i. Subtract 3 from both sides: \(-5 - 3 \leq -4y\), leading to \(-8 \leq -4y\).ii. Divide both sides by \(-4\), remembering to reverse the inequality: \(\frac{-8}{-4} \geq y\), leading to \(2 \geq y\).
03

Solve the Second Inequality

Solve \(-4y + 3 < 7\):i. Subtract 3 from both sides: \(-4y < 4\).ii. Divide both sides by \(-4\), remembering to reverse the inequality: \(\frac{4}{-4} > y\), leading to \(-1 > y\) or equivalently \(y < -1\).
04

Combine the Solutions

Combine the two results from Step 2 and Step 3: \(2 \geq y\) and \(y < -1\). This means the solution is \(-1 < y \leq 2\).
05

Express the Final Answer

The solution to the inequality \(-5 \leq -4y + 3 < 7\) is \(-1 < y \leq 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

compound inequalities
When dealing with compound inequalities, you are working with two inequalities at once. For example, the inequality \(-5 \leq-4 y+3<7\) is a compound inequality. This means you need to solve two separate inequalities and find the solution that satisfies both. The given inequality can be broken into two parts:
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Most popular questions from this chapter

Summarize the properties of equality in one sentence.

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