1.1.1 Solve the Linear Inequality

We are given the inequality:
$-6 < 2x - 4 \leq 2$ We solve it step by step.

Step 1: Add 4 to all sides

$-6 + 4 < 2x - 4 + 4 \leq 2 + 4$ $-2 < 2x \leq 6$

Step 2: Divide by 2

$\frac{-2}{2} < \frac{2x}{2} \leq \frac{6}{2}$ $-1 < x \leq 3$

Thus, the solution is:
$(-1, 3]$ This means $x$ is greater than $-1$ but less than or equal to $3$.

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1.1.2 Represent on a Number Line

On a number line:

• Use an open circle at $x = -1$ (since $-1$ is not included).
• Use a closed circle at $x = 3$ (since $3$ is included).
• Shade the region between them.

Here’s a rough representation:

----○================●----
   -1                3

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Would you like a more detailed explanation or a drawn number line?

Related Questions:

1. How do you solve and graph compound inequalities?
2. What are the differences between strict inequalities ($<, >$) and inclusive ones ($\leq, \geq$)?
3. How do you represent inequalities using interval notation?
4. What happens if we multiply or divide by a negative number in an inequality?
5. How can inequalities be used in real-life situations?

Tip:

When solving inequalities, always remember that multiplying or dividing by a negative number reverses the inequality sign.