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Chapter 2: Problem 48

For exercises 37-52, (a) solve. (b) use a number line graph to represent the solution. (c) check the direction of the inequality sign. $$ 3(4 x-1) \leq 9(x-3) $$

Short Answer

Expert verified
The solution is \( x \leq -8 \).

Step by step solution

01

Distribute the constants

First, expand both sides of the inequality by distributing the constants inside the parentheses. \[ 3(4x - 1) = 12x - 3 \] \[ 9(x - 3) = 9x - 27 \]
02

Combine like terms

Next, combine like terms to simplify the inequality. \[ 12x - 3 \leq 9x - 27 \]
03

Isolate the variable term

Move all terms involving \(x\) to one side and constants to the other by subtracting \(9x\) from both sides. \[ 12x - 9x - 3 \leq -27 \] This simplifies to: \[ 3x - 3 \leq -27 \]
04

Solve for the variable

Isolate \(x\) by first adding 3 to both sides: \[ 3x - 3 + 3 \leq -27 + 3 \] This simplifies to: \[ 3x \leq -24 \] Now, divide both sides by 3: \[ x \leq -8 \]
05

Graph the solution on a number line

Draw a number line and shade the region to the left of \(x = -8\) to represent all values less than or equal to \(-8\). Include a closed circle on \(-8\) to indicate that \(-8\) is part of the solution.
06

Check the inequality direction

Select a test point less than or equal to \(-8\) and substitute it back into the original inequality to ensure the solution is correct. For example, using \(x = -9\): \[ 3(4(-9) - 1) \leq 9(-9 - 3) \] This simplifies to: \[ 3(-36 - 1) \leq 9(-12) \] \[ 3(-37) \leq -108 \] \[ -111 \leq -108 \] Since the statement is true, the solution \(x \leq -8\) is correct.

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Key Concepts

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

Distributive Property
To further illustrate, let's consider another example: \[ 2(a + 5) \] This expands to: \[ 2 \times a + 2 \times 5 = 2a + 10 \] Practice solving these types of problems to become comfortable with distributing constants.
Combining Like Terms
Always combine terms with the same variables and exponents. For example: \[ 5y + 3y = 8y \] But you cannot combine these: \[ 2x + 3y \] because they have different variables.
Isolating Variables
Always perform the same operation on both sides of the equation. If you add, subtract, multiply, or divide one side by a number, do the same to the other side. This keeps the equation balanced. For example: \[ x/4 = 8 \] Multiply both sides by 4 to isolate x: \[ x = 32 \]
Number Line Representation
Double-check your shaded area and point to ensure they correctly represent your solution. The closed circle indicates that -8 is included in the values that satisfy the inequality: \[ x \] Using a test point can help confirm the correctness of your solution. For instance, substitute -9 into the original inequality to verify.

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Most popular questions from this chapter

For exercises 65-86, (a) solve. (b) check. $$ \frac{2}{3}(n-1)+\frac{1}{6}=n-\frac{1}{2} $$

For problems \(89-92\), do the arithmetic with a calculator. The area \(A\) of a rectangular playground is \(28,800 \mathrm{ft}^{2}\). The length \(L\) is \(180 \mathrm{ft}\). Solve the formula \(A=L W\) for \(L\), and use it to find the width \(W\) of the playground.

For exercises 1-28, solve the equation for \(y\). Write the equation to match the pattern \(y=m x+b\). $$ -5 x=\frac{1}{6} y $$

For exercises 1-72, (a) solve. (b) check. $$ 4=-5-x $$

Describe the difference in the solution of \(x<9\) and the solution of \(x \leq 9\).
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