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Chapter 3: Problem 46

Solve each inequality. $$1

Short Answer

Expert verified
The solution is \(-4 < x < 4\).

Step by step solution

01

Break Down the Compound Inequality

The compound inequality given is \(1 < x + 5 < 9\). This can be split into two inequalities: 1. \(1 < x + 5\)2. \(x + 5 < 9\).
02

Solve the First Inequality

The first inequality is: \(1 < x + 5\). Subtract 5 from both sides to isolate x: \(1 - 5 < x + 5 - 5\) This simplifies to: \(-4 < x\).
03

Solve the Second Inequality

The second inequality is: \(x + 5 < 9\). Subtract 5 from both sides to isolate x: \(x + 5 - 5 < 9 - 5\) This simplifies to: \(x < 4\).
04

Combine the Results

Now we combine the results of the two inequalities \(-4 < x\) and \(x < 4\) This gives us the solution: \(-4 < x < 4\).

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Key Concepts

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

headline of the respective core concept
Inequalities are mathematical expressions that show the relationship between two values where one is not equal to the other. They use signs like greater than (>) and less than (<). For example, if we say 'x > 2', we're stating that x is any number greater than 2. There are different types of inequalities, but they all help us understand how values compare to one another.
headline of the respective core concept
Solving inequalities is similar to solving equations. Our goal is to isolate the variable, just like in equations, but we also need to be mindful of the inequality sign. Here's a general method:
  • Step 1: Start by isolating the variable on one side of the inequality.
  • Step 2: Perform the same operation on both sides, just like balancing an equation.
  • Step 3: If you multiply or divide by a negative number, reverse the inequality sign.
For example, to solve for x in the inequality 1 < x + 5:
1. Subtract 5 from both sides to get -4 < x.
headline of the respective core concept
Compound inequalities involve two separate inequalities that are combined into one statement using 'and' or 'or'. In an 'and' compound inequality, both conditions must be true simultaneously. For example, in the problem given, we have 1 < x + 5 < 9. This can be broken down into two separate inequalities:
  • First: 1 < x + 5
  • Second: x + 5 < 9
After solving both, we combine the results to get -4 < x < 4. This means x must be greater than -4 and less than 4 at the same time. Compound inequalities help us understand ranges that a variable can fall within.

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Most popular questions from this chapter

Summarize the properties of equality in one sentence.

Evaluate the given expression. $$\frac{-15-5}{-2(-5)(-2)}$$

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