https://file.aiquickdraw.com/mathbot/file/1726404120112wst7nu1o.jpg

The image presents two inequalities to solve and represent graphically on a number line:

1. \( 2(x - 1) < x + 5 \)
2. \( 3(x + 2) > 2 - x \)

### Step 1: Solve each inequality separately.

#### Inequality 1:
\[
2(x - 1) < x + 5
\]
Expand both sides:
\[
2x - 2 < x + 5
\]
Subtract \(x\) from both sides:
\[
x - 2 < 5
\]
Add 2 to both sides:
\[
x < 7
\]

So, the solution for the first inequality is:
\[
x < 7
\]

#### Inequality 2:
\[
3(x + 2) > 2 - x
\]
Expand both sides:
\[
3x + 6 > 2 - x
\]
Add \(x\) to both sides:
\[
4x + 6 > 2
\]
Subtract 6 from both sides:
\[
4x > -4
\]
Divide both sides by 4:
\[
x > -1
\]

So, the solution for the second inequality is:
\[
x > -1
\]

### Step 2: Combine the solutions
We have:
1. \(x < 7\)
2. \(x > -1\)

The combined solution is:
\[
-1 < x < 7
\]

### Step 3: Represent the solution graphically on a number line
On the number line, you would place an open circle at \(x = -1\) and \(x = 7\) because these values are not included (strict inequalities), and shade the region between them to indicate the solution set for \(x\).

---

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How do we interpret open and closed circles in inequalities on a number line?
2. What would happen if the inequalities were non-strict, like \(x \leq 7\)?
3. How do we solve compound inequalities that involve "and" or "or"?
4. How would this process change if we had quadratic inequalities instead of linear?
5. Can inequalities be represented in any other way apart from a number line?

**Tip:** When solving inequalities, always be cautious about the direction of the inequality sign, especially when multiplying or dividing both sides by a negative number (it reverses the inequality).

Solve the inequalities and represent the solution graphically on a number line: 2(x - 1) < x + 5, 3(x + 2) > 2 - x

Learn how to solve linear inequalities and graphically represent solutions on a number line. This guide covers the step-by-step process for inequalities 2(x - 1) < x + 5 and 3(x + 2) > 2 - x. Perfect for students in grades 8-10 looking to master inequality solutions.

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