Answer :
To solve the inequalities [tex]\(3x < x + 4\)[/tex] and [tex]\(\frac{1}{2}(4x - 6) > x - 2\)[/tex], we will proceed step by step for each inequality and then find the intersection of the solutions.
### Step 1: Solve [tex]\(3x < x + 4\)[/tex]
1. Subtract [tex]\(x\)[/tex] from both sides to simplify:
[tex]\[ 3x - x < x + 4 - x \][/tex]
[tex]\[ 2x < 4 \][/tex]
2. Divide both sides by 2:
[tex]\[ \frac{2x}{2} < \frac{4}{2} \][/tex]
[tex]\[ x < 2 \][/tex]
The solution to the inequality [tex]\(3x < x + 4\)[/tex] is [tex]\(x < 2\)[/tex].
### Step 2: Solve [tex]\(\frac{1}{2}(4x - 6) > x - 2\)[/tex]
1. Distribute [tex]\(\frac{1}{2}\)[/tex] inside the parentheses:
[tex]\[ \frac{1}{2} \cdot 4x - \frac{1}{2} \cdot 6 > x - 2 \][/tex]
[tex]\[ 2x - 3 > x - 2 \][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides to simplify:
[tex]\[ 2x - x - 3 > x - x - 2 \][/tex]
[tex]\[ x - 3 > -2 \][/tex]
3. Add 3 to both sides:
[tex]\[ x - 3 + 3 > -2 + 3 \][/tex]
[tex]\[ x > 1 \][/tex]
The solution to the inequality [tex]\(\frac{1}{2}(4x - 6) > x - 2\)[/tex] is [tex]\(x > 1\)[/tex].
### Step 3: Intersection of the solutions
We need to find the values of [tex]\(x\)[/tex] that satisfy both [tex]\(x < 2\)[/tex] and [tex]\(x > 1\)[/tex].
The intersection of the two inequalities is:
[tex]\[ 1 < x < 2 \][/tex]
### Final Answer
The solution to the inequalities [tex]\(3x < x + 4\)[/tex] and [tex]\(\frac{1}{2}(4x - 6) > x - 2\)[/tex] is [tex]\(1 < x < 2\)[/tex].
### Number Line Representation
On a number line, this solution is represented as:
[tex]\[ (1, 2) \][/tex]
```
1 2
------(===)-----
```
The open parentheses and segment indicate that [tex]\(x\)[/tex] is greater than 1 and less than 2, but not including the endpoints 1 and 2 themselves.
### Step 1: Solve [tex]\(3x < x + 4\)[/tex]
1. Subtract [tex]\(x\)[/tex] from both sides to simplify:
[tex]\[ 3x - x < x + 4 - x \][/tex]
[tex]\[ 2x < 4 \][/tex]
2. Divide both sides by 2:
[tex]\[ \frac{2x}{2} < \frac{4}{2} \][/tex]
[tex]\[ x < 2 \][/tex]
The solution to the inequality [tex]\(3x < x + 4\)[/tex] is [tex]\(x < 2\)[/tex].
### Step 2: Solve [tex]\(\frac{1}{2}(4x - 6) > x - 2\)[/tex]
1. Distribute [tex]\(\frac{1}{2}\)[/tex] inside the parentheses:
[tex]\[ \frac{1}{2} \cdot 4x - \frac{1}{2} \cdot 6 > x - 2 \][/tex]
[tex]\[ 2x - 3 > x - 2 \][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides to simplify:
[tex]\[ 2x - x - 3 > x - x - 2 \][/tex]
[tex]\[ x - 3 > -2 \][/tex]
3. Add 3 to both sides:
[tex]\[ x - 3 + 3 > -2 + 3 \][/tex]
[tex]\[ x > 1 \][/tex]
The solution to the inequality [tex]\(\frac{1}{2}(4x - 6) > x - 2\)[/tex] is [tex]\(x > 1\)[/tex].
### Step 3: Intersection of the solutions
We need to find the values of [tex]\(x\)[/tex] that satisfy both [tex]\(x < 2\)[/tex] and [tex]\(x > 1\)[/tex].
The intersection of the two inequalities is:
[tex]\[ 1 < x < 2 \][/tex]
### Final Answer
The solution to the inequalities [tex]\(3x < x + 4\)[/tex] and [tex]\(\frac{1}{2}(4x - 6) > x - 2\)[/tex] is [tex]\(1 < x < 2\)[/tex].
### Number Line Representation
On a number line, this solution is represented as:
[tex]\[ (1, 2) \][/tex]
```
1 2
------(===)-----
```
The open parentheses and segment indicate that [tex]\(x\)[/tex] is greater than 1 and less than 2, but not including the endpoints 1 and 2 themselves.
