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Chapter 5: Problem 7

Solve the following linear inequalities. $$ 18 \geq 4(2 x-3)-9 x $$

Short Answer

Expert verified
Question: Solve the linear inequality and express the solution set: \(18 \geq 4(2x - 3) - 9x\) Answer: \(x \geq -30\)

Step by step solution

01

Distribute the 4 through the parenthesis

To simplify the inequality, we need to distribute the 4 to both terms inside the parentheses. $$ 18 \geq 4(2x - 3) - 9x $$ Becomes: $$ 18 \geq 8x - 12 - 9x $$
02

Combine like terms

Combine the x terms (8x and -9x) on the right-hand side of the inequality. $$ 18 \geq 8x - 9x - 12 $$ Becomes: $$ 18 \geq -1x - 12 $$
03

Add 12 to both sides of the inequality

To isolate the x term, let's add 12 to both sides of the inequality. $$ 18 + 12 \geq -1x - 12 + 12 $$ Simplifying, we get: $$ 30 \geq -1x $$
04

Divide both sides by -1

Now, we need to get x by itself by dividing both sides of the inequality by -1. Remember, when you divide or multiply an inequality by a negative number, the inequality changes direction. $$ \frac{30}{-1} \leq \frac{-1x}{-1} $$ Simplifying, we get: $$ -30 \leq x $$
05

Write the final answer

The solution is written as: $$ x \geq -30 $$ Hence, the solution for the given linear inequality is \(x \geq -30\).

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Key Concepts

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

Simplify Inequality
When solving linear inequalities, the first crucial step is to simplify the inequality. This process involves distributing any multiplication across terms within parentheses and combining any constants. Let's consider the example given:
\( 18 \geq 4(2x - 3) - 9x \).
Here, to simplify, we distribute the 4 into the parentheses, resulting in \( 8x - 12 \). Simplification makes the inequality easier to work with by reducing it down to its most basic form. Remember, the aim here is to create a clearer path to isolate the variable, which is the focus of the subsequent steps.
Combine Like Terms
With the inequality simplified, the next concept is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In the example, \( 8x \) and \( -9x \) are like terms. Combining these gives us \( -1x \), transforming the equation to \( 18 \geq -1x - 12 \). This action consolidates the variable terms, thus setting the stage for isolating the variable. Simplifying by combining like terms paves the way for a clearer visualization of what you are solving for, and it's essential for reaching the most straightforward form of the inequality.
Isolate Variable
Isolating the variable is the final step in solving linear inequalities. In our example, we aim to isolate \( x \). To do this, we want all terms containing \( x \) on one side of the inequality and the constants on the other. After combining like terms, we were left with \( 30 \geq -1x \). To isolate \( x \), we must divide both sides by \( -1 \) 鈥� but crucially, this action reverses the inequality, changing \( \geq \) to \( \leq \) to maintain the inequality's truth. Therefore, isolating \( x \) in the example gives us the solution of \( x \geq -30 \). By mastering variable isolation, you develop a method to solve for the unknown in any linear inequality.

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Most popular questions from this chapter

Solve the inequalities. $$ -x-5<3 x-11 $$

Find the solution. A television commercial advertises that a certain type of light bulb will last, on the average, 200 hours longer than three times the life of another type of bulb. If consumer tests show that the advertised bulb lasts 4700 hours, how many hours must the other type of bulb last for the advertiser's claim to be valid?

Write "identity," "contradiction," or "conditional." If you can, find the solution by making an educated guess based on your knowledge of arithmetic. $$ x+1=10 $$

Write "identity," "contradiction," or "conditional." If you can, find the solution by making an educated guess based on your knowledge of arithmetic. $$ x+4=x-3 $$

Translate the phrases or sentences to mathematical expressions or equations. Two fifths of a number minus five.
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