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Chapter 2: Problem 112

Graph the solution set, and write it using interval notation \(-4<2(x+1) \leq 6\)

Short Answer

Expert verified
The solution in interval notation is \((-3, 2]\).

Step by step solution

01

- Simplify the inequality

First, simplify the inequality by distributing the 2 on the right side: \(-4 < 2(x + 1) \leq 6\) becomes \(-4 < 2x + 2 \leq 6\).
02

- Isolate the variable term

Next, subtract 2 from all parts of the inequality to get all terms involving x isolated on one side: \(-4 - 2 < 2x + 2 - 2 \leq 6 - 2\) Simplifying, we get: \(-6 < 2x \leq 4\).
03

- Solve for x

Divide every part of the inequality by 2 to solve for x: \(\frac{-6}{2} < \frac{2x}{2} \leq \frac{4}{2}\) Simplifying, we get: \(-3 < x \leq 2\).
04

- Write the solution in interval notation

The solution in interval notation is written as the set of x values that satisfy the inequality: \((-3, 2]\).
05

- Graph the solution set

Graph the solution set on a number line. Use an open circle at -3 to indicate that -3 is not included in the solution set, and a closed circle at 2 to indicate that 2 is included. Shade the region in between. The graph looks like this:Open circle on -3, closed circle on 2, and shading between them.

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Key Concepts

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

Interval Notation
Interval notation is a simple way of expressing a range of values. Instead of writing all the possible values individually, you can write a compact representation. For example, \((-3, 2]\) means that x can be any value between -3 and 2, but -3 itself is not included, while 2 is.

In interval notation:
  • Round brackets \(( )\) indicate that the endpoint is not included.
  • Square brackets \[ ]\] indicate that the endpoint is included.
The interval notation \((-3, 2]\) captures our solution perfectly.
This notation makes it clear and easy to understand the range of values that satisfy our inequality.
Inequality Graph
Graphing inequalities on a number line helps visualize the set of possible solutions. In this case, our solution set is \(-3 < x \leq 2\).
This interval means that x is greater than -3 but less than or equal to 2.
Here鈥檚 how to graph the solution set:
  • Draw a number line.
  • Place an open circle at -3 to show that -3 is not included.
  • Place a closed circle at 2 to show that 2 is included.
  • Shade the region between -3 and 2.
These visual cues make it easy to understand which values are part of the solution set.
Variable Isolation
Isolating the variable is a critical step in solving inequalities. It involves simplifying the inequality until the variable is alone on one side. Here鈥檚 how we did it:

First, we started with the inequality \(-4< 2(x + 1) \leq 6\).
  • We distributed the 2: \(-4 < 2x + 2 \leq 6\).
  • Next, we subtracted 2 from each part of the inequality: \(-6 < 2x \leq 4\).
  • Finally, we divided each part by 2: \(-3 < x \leq 2\).
Each step maintains the balance of the inequality, ensuring that we find the correct solution.
Solving Inequalities
Solving inequalities is similar to solving equations, but with an added focus on the inequality signs. Follow these key steps:
  • Start with the given inequality: \(-4< 2(x + 1) \leq 6\).
  • Distribute any numbers outside parentheses: \(-4 < 2x + 2 \leq 6\).
  • Subtract or add terms to isolate the variable: \(-6 < 2x \leq 4\).
  • Divide or multiply to solve for the variable: \(-3 < x \leq 2\).
  • Write the solution using interval notation.
Pay attention to the inequality symbols. If you multiply or divide by a negative number, the inequality sign flips.
This isn鈥檛 needed here, but it鈥檚 crucial in other problems.

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Most popular questions from this chapter

In the 2017 regular baseball season, the Cleveland Indians won 18 fewer than twice as many games as they lost. They played 162 regular-season games. How many wins and losses did the team have? (Data from www.MLB.com)

Solve each formula for the specified variable. \(M=C(1+r)\) for \(r\)

Find two consecutive even integers such that the lesser added to three times the greater gives a sum of \(46 .\)

When the lesser of two consecutive integers is added to three times the greater, the result is \(43 .\) Find the integers.

Graph the solution set, and write it using interval notation \(6 \leq 3(x-1)<18\)
See all solutions

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