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Magazine Chapter 4: Problem 66
Solve and graph each solution set. Write the answer using both set-builder notation and interval notation. $$ 7 \geq g(x) \geq-2, \text { where } g(x)=3 x-5 $$
Short Answer
Expert verified
[1, 4] or \( \{ x | 1 \leq x \leq 4 \} \)
Step by step solution
01
Set Up the Inequalities
Given the compound inequality: \[ 7 \geq g(x) \geq -2 \] Substituting for \( g(x) \): \[ 7 \geq 3x - 5 \geq -2 \]
02
Separate the Inequalities
Break down the compound inequality into two simpler inequalities: 1) \[ 7 \geq 3x - 5 \] 2) \[ 3x - 5 \geq -2 \]
03
Solve the First Inequality
Solve \( 7 \geq 3x - 5 \): 1) Add 5 to both sides: \[ 7 + 5 \geq 3x \] 2) Simplify: \[ 12 \geq 3x \] 3) Divide both sides by 3: \[ 4 \geq x \] 4) Rewrite as: \[ x \leq 4 \]
04
Solve the Second Inequality
Solve \( 3x - 5 \geq -2 \): 1) Add 5 to both sides: \[ 3x - 5 + 5 \geq -2 + 5 \] 2) Simplify: \[ 3x \geq 3 \] 3) Divide both sides by 3: \[ x \geq 1 \]
05
Combine the Solutions
Combine the two separate inequalities: \[ 1 \leq x \leq 4 \]
06
Write Solution in Set-Builder Notation
The solution set in set-builder notation is: \[ \{ x | 1 \leq x \leq 4 \} \]
07
Write Solution in Interval Notation
The solution set in interval notation is: \[ [1, 4] \]
08
Graph the Solution
Graph the interval on a number line: Draw a number line, and shade the region between 1 and 4, including the endpoints since both are valid: ---|---鈥�---鈥�---------|----鈭�---1---2---3---4---鈭�
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Set-Builder Notation
Set-builder notation is a concise way of describing a set by indicating the properties that its members must satisfy. For example, the set-builder notation for the solution to the inequality from the exercise is written as \( \{ x \ | \ 1 \leq x \leq 4 \} \). This can be read as 'the set of all x such that x is greater than or equal to 1 and less than or equal to 4'. In set-builder notation:
- The vertical bar \( \ | \) means 'such that'.
- It describes all values that make the statement true.
- It is a handy way to express unlimited or large sets.
Interval Notation
Interval notation is a way to describe the set of solutions to an inequality. It involves writing the interval within brackets. For the given problem, the solution in interval notation is \( [1, 4] \). Here's how it works:
- The brackets \( [ \ \ ] \) indicate that the endpoints are included (closed interval).
- If parentheses \( ( \ \ ) \) were used, it would indicate the endpoints are not included (open interval).
- For example, \( [1, 4] \) means all numbers between 1 and 4, including 1 and 4.
Solving Inequalities
Solving inequalities involves finding the values of a variable that make the inequality true. For the exercise, we handled the compound inequality \( 7 \geq g(x) \geq-2 \). Here's a step-by-step breakdown of the process:
- First, substitute \( g(x) = 3x - 5 \) into the inequality: \( 7 \geq 3x - 5 \geq -2 \).
- Then, separate into two inequalities: \( 7 \geq 3x - 5 \) and \( 3x - 5 \geq -2 \).
- Solve each inequality individually:
- Add 5 to both sides of \( 7 \geq 3x - 5 \) to get \( 12 \geq 3x \), then divide by 3: \( 4 \geq x \) (or \( x \leq 4 \).
- Add 5 to both sides of \( 3x - 5 \geq -2 \) to get \( 3x \geq 3 \), then divide by 3: \( x \geq 1 \).
Graphing Solution Sets
Graphing the solution sets of inequalities on a number line provides a visual representation of the range of solutions. Here's how to graph the solution \( 1 \leq x \leq 4 \) on a number line:
- Draw a horizontal line representing the number line.
- Mark the points corresponding to the endpoints (1 and 4).
- Use a solid dot (鈥�) to indicate that 1 and 4 are included in the solution (closed interval).
- Shade the region between these points to show all valid values for x.
