Math Problem Statement

Solve the following inequality, and find the interval notation of the solution set: −3<2x+1≤5

Solution

Let's solve the inequality step by step:

We are given the inequality:

3<2x+15-3 < 2x + 1 \leq 5

Step 1: Break it into two parts

We can break this compound inequality into two separate inequalities and solve each part:

  1. 3<2x+1-3 < 2x + 1
  2. 2x+152x + 1 \leq 5

Step 2: Solve the first inequality 3<2x+1-3 < 2x + 1

Subtract 1 from both sides:

31<2x-3 - 1 < 2x 4<2x-4 < 2x

Now, divide both sides by 2:

42<x\frac{-4}{2} < x x>2x > -2

Step 3: Solve the second inequality 2x+152x + 1 \leq 5

Subtract 1 from both sides:

2x512x \leq 5 - 1 2x42x \leq 4

Now, divide both sides by 2:

x42x \leq \frac{4}{2} x2x \leq 2

Step 4: Combine the two results

We now have:

2<x2-2 < x \leq 2

Step 5: Write the solution in interval notation

The solution set in interval notation is:

(2,2](-2, 2]

This means that xx is strictly greater than 2-2 but less than or equal to 22.

Do you want more details or have any questions?

Here are 5 related questions:

  1. How do you solve inequalities involving absolute values?
  2. What is the difference between strict inequalities and non-strict inequalities?
  3. Can you explain how to solve inequalities involving quadratic expressions?
  4. How does interval notation differ from set notation?
  5. What happens when an inequality is multiplied or divided by a negative number?

Tip: When solving inequalities, remember that multiplying or dividing both sides by a negative number reverses the inequality sign.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Interval Notation

Formulas

Basic inequality manipulation (adding, subtracting, multiplying, and dividing)

Theorems

Properties of inequalities

Suitable Grade Level

Grades 7-9