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Chapter 3: Problem 32

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ -4 x<24 $$

Short Answer

Expert verified
The solution is \(x > -6\).

Step by step solution

01

Understand the Inequality

The inequality given is \(-4x < 24\). Our goal is to find the values of \(x\) that satisfy this inequality.
02

Isolate the Variable

In order to isolate \(x\), we need to divide both sides of the inequality by \(-4\). Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. Therefore, when we divide both sides by \(-4\), the inequality becomes:\[x > \frac{24}{-4}\]
03

Simplify the Expression

Simplify the right-hand side of the inequality. Compute \(\frac{24}{-4}\) to find:\[x > -6\]
04

State the Solution

The solution to the inequality \(-4x < 24\) is all values of \(x\) that satisfy \(x > -6\).

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Key Concepts

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

Isolating the Variable
When solving inequalities like \(-4x < 24\), the main goal is to get the variable on one side of the inequality by itself. This is called isolating the variable. First, we need to do the same operation on both sides of the inequality to maintain balance.
In our example, the variable \(x\) is attached to the number \(-4\). This means we have to divide both sides of the inequality by \(-4\) to "undo" the multiplication:
  • This effectively cancels out the \(-4\) next to the \(x\), resulting in \(x\) being alone.
  • On the right side, you will perform the division operation to simplify the inequality.

These steps are crucial in helping us progress towards the solution.
Reversing the Inequality Sign
Dividing or multiplying both sides of an inequality by a negative number requires you to reverse the inequality sign. This is a fundamental rule of inequalities and is important to maintain the equivalency of the inequality statement.
In the provided inequality \(-4x < 24\), after dividing both sides by \(-4\), we reverse the inequality sign from '<' to '>' resulting in:
  • \(x > \frac{24}{-4}\)

This step ensures the inequality expresses the correct relationship between the two sides after the operation involving a negative number.
Simplifying Expressions
In the context of solving the inequality \(x > \frac{24}{-4}\), we need to perform the division to simplify the expression. Simplifying expressions involves breaking down a mathematical expression to its simplest form.
For this inequality:
  • Calculate \(\frac{24}{-4}\) which simplifies to \(-6\).
  • As a result, we obtain \(x > -6\).

Simplifying expressions allows you to clearly see the solution to the inequality. It also helps in understanding the range of values that the variable can take. In this case, \(x\) can be any number greater than \(-6\), effectively solving the inequality.

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