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Chapter 2: Problem 108

Graph the solution set, and write it using interval notation \(-7 \leq 3 x-4 \leq 8\)

Short Answer

Expert verified
The solution set is \( [-1, 4] \).

Step by step solution

01

Isolate the linear expression

First, isolate the linear expression in the middle by adding 4 to all parts of the inequality:\[ -7 + 4 \leq 3x - 4 + 4 \leq 8 + 4 \]This simplifies to:\[ -3 \leq 3x \leq 12 \]
02

Solve for x

Now, solve for \(x\) by dividing all parts of the inequality by 3:\[ \frac{-3}{3} \leq \frac{3x}{3} \leq \frac{12}{3} \]This simplifies to:\[ -1 \leq x \leq 4 \]
03

Graph the solution set

To graph the solution set on a number line, draw a line with points from -1 to 4. Include filled-in circles at -1 and 4 because \(x\) is greater than or equal to -1 and less than or equal to 4, which means -1 and 4 are included in the solution set.
04

Write the interval notation

In interval notation, the solution set is written as:\[ [-1, 4] \]

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Key Concepts

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

graphing inequalities
Graphing inequalities helps us visually understand the range of values that satisfy an inequality.
For the inequality \( -1 \leq x \leq 4 \), we need to graph this on a number line:
  • Start by drawing a horizontal line.
  • Mark points at -1 and 4 on the line.
  • Place filled-in circles at -1 and 4 since the inequality includes both these values.
  • Shade the region between -1 and 4 to show all the numbers that satisfy the inequality.
This shaded region represents all the values of \( x \) that make the original inequality true.
Visualizing inequalities this way makes it clearer which numbers satisfy the conditions.
solving linear inequalities
Solving linear inequalities involves similar steps as solving regular equations, with a few additional rules.
Here's a quick guide:
  • Step 1: Isolate the linear expression. In the given problem, we start with \( -7 \leq 3 x-4 \leq 8 \). Add 4 to all parts to get: \[ -3 \leq 3x \leq 12 \]
  • Step 2: Solve for the variable by dividing all parts by the coefficient of x. Here, divide by 3: \[ \frac{-3}{3} \leq \frac{3x}{3} \leq \frac{12}{3} \], which simplifies to \[ -1 \leq x \leq 4 \]
It's important to pay attention to the direction of the inequalities, especially when multiplying or dividing by negative numbers, as this reverses the inequality sign.
However, in our problem, dividing by a positive number doesn't change the inequality direction.
number lines
Number lines are useful tools for graphing inequalities and understanding the range of possible values for variables.
To effectively use a number line:
  • Draw a horizontal line and label it with evenly spaced numbers.
  • For inequalities that include the boundary values (like \( x \leq 4 \)), use filled-in circles at those points.
  • Shade between the points to show all numbers that fit the inequality. For \( -1 \leq x \leq 4 \), shade the region from -1 to 4.
Number lines help you easily see the solution set of an inequality.
For \([-1, 4]\) in interval notation, the filled-in circles and shading directly translate to including the endpoints (-1 and 4).
This visual tool simplifies complex concepts and helps you check your solutions.

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