Math Problem Statement

1.1 Given the following linear inequality: − 6 < 2𝑥 − 4 ≤ 2 1.1.1 Solve the linear inequality. (2) 1.1.2 Represent your solution on a number line. (1)

Solution

1.1.1 Solve the linear inequality:

Given the inequality:
6<2x42-6 < 2x - 4 \leq 2

Step 1: Solve the first part of the inequality:
6<2x4-6 < 2x - 4 Add 4 to both sides: 6+4<2x2<2x-6 + 4 < 2x \quad \Rightarrow \quad -2 < 2x Now, divide both sides by 2: 22<2x21<x\frac{-2}{2} < \frac{2x}{2} \quad \Rightarrow \quad -1 < x So, the first part simplifies to: x>1x > -1

Step 2: Solve the second part of the inequality:
2x422x - 4 \leq 2 Add 4 to both sides: 2x2+42x62x \leq 2 + 4 \quad \Rightarrow \quad 2x \leq 6 Now, divide both sides by 2: 2x262x3\frac{2x}{2} \leq \frac{6}{2} \quad \Rightarrow \quad x \leq 3

Step 3: Combine both parts: x>1andx3x > -1 \quad \text{and} \quad x \leq 3 So the solution is: 1<x3-1 < x \leq 3

1.1.2 Represent your solution on a number line:

The solution 1<x3-1 < x \leq 3 is represented as:

  • An open circle at 1-1 (because xx cannot equal 1-1),
  • A filled circle at 33 (because xx can equal 33),
  • A shaded region between 1-1 and 33 to indicate all values of xx in this range.
\quad -1 \quad 0 \quad 1 \quad 2 \quad 3$$ Would you like further clarification on any step? Let me know if you'd like to explore related questions! Here are 5 related questions: 1. How would you solve a similar inequality involving fractions or decimals? 2. What happens if the inequality is reversed, like $$2x - 4 \geq -6$$? 3. How would you graph a solution to an inequality involving absolute values? 4. How can you solve a compound inequality with "and" and "or"? 5. Can you solve the inequality $$3(x - 2) \geq 5x - 4$$? **Tip:** Always be careful when dividing or multiplying by negative numbers in inequalities, as this reverses the direction of the inequality!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Inequalities
Number Line Representation

Formulas

Solving compound inequalities
Properties of inequalities

Theorems

Addition and multiplication properties of inequalities

Suitable Grade Level

Grades 8-10