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Chapter 4: Problem 21

Solve each inequality. $$-3 \leq h-2 \leq 1$$

Short Answer

Expert verified
The solution is \(-1 \leq h \leq 3\).

Step by step solution

01

Understand the Compound Inequality

The inequality given is a compound inequality. This means you need to solve two separate inequalities: \(-3 \leq h - 2 \quad\text{and}\quad h - 2 \leq 1\).
02

Solve the First Inequality

First, solve \(-3 \leq h - 2\). Add 2 to both sides of the inequality to isolate \(h\): \(-3 + 2 \leq h - 2 + 2\) This simplifies to: \(-1 \leq h\).
03

Solve the Second Inequality

Next, solve \(h - 2 \leq 1\). Add 2 to both sides of the inequality to isolate \(h\): \(h - 2 + 2 \leq 1 + 2\) This simplifies to: \(h \leq 3\).
04

Combine the Solutions

Now combine the results from Step 2 and Step 3: \(-1 \leq h \quad\text{and}\quad h \leq 3\) This can be written as: \(-1 \leq h \leq 3\).

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Key Concepts

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

compound inequality
A compound inequality is a combination of two or more inequalities joined by 'and' or 'or'. In this exercise, we have a compound inequality joined by 'and': \(-3 \leq h-2 \leq 1\). This means we need to find the values of \(h\) that satisfy both inequalities simultaneously. Specifically, we need \(-3 \leq h-2\) and \(h-2 \leq 1\) to be true at the same time. Compound inequalities allow us to describe a range of values that a variable can take. In this case, we will end up finding that \(h\) lies between two numbers.
isolating variables
To solve an inequality, especially a compound inequality, one common method is isolating the variable. Isolating the variable means getting the variable (in this case, \(h\)) by itself on one side of the inequality sign.

For \(-3 \leq h-2\), we first need to get rid of the \(-2\) next to \(h\). We do this by adding 2 to both sides of the inequality:
\(-3 + 2 \leq h - 2 + 2\).
This simplifies to:
\(-1 \leq h\).

Similarly, for \(h-2 \leq 1\), we add 2 to both sides to isolate \(h\):\
\(h - 2 + 2 \leq 1 + 2\).
This simplifies to:
\(h \leq 3\). By isolating \(h\) in both inequalities, we can clearly see the range of values that \(h\) can take.
inequality solution steps
Solving inequalities often involves multiple steps to ensure we find the correct range of solutions. Here are the crucial steps we used to solve the given compound inequality \(-3 \leq h-2 \leq 1\):

  • Step 1: Understand that the original inequality is a compound inequality, meaning we need to solve each part separately.

  • Step 2: Solve the first inequality \(-3 \leq h-2\). Add 2 to both sides to isolate \(h\), getting \(-1 \leq h\).

  • Step 3: Solve the second inequality \(h-2 \leq 1\). Similarly, add 2 to both sides to isolate \(h\), getting \(h \leq 3\).

  • Step 4: Combine the results from the two inequalities to get the final solution: \(-1 \leq h \leq 3\). This tells us that \(h\) takes values between -1 and 3 inclusive.

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