Answer :
To solve and graph the given compound inequality, we will address each inequality individually and then combine the solutions obtained.
### Inequality 1: [tex]\(2(2x - 1) + 7 < 13\)[/tex]
1. Distribute the 2 on the left side:
[tex]\[ 2(2x - 1) + 7 < 13 \][/tex]
[tex]\[ 4x - 2 + 7 < 13 \][/tex]
2. Combine like terms:
[tex]\[ 4x + 5 < 13 \][/tex]
3. Isolate the variable [tex]\(x\)[/tex]:
- Subtract 5 from both sides:
[tex]\[ 4x < 8 \][/tex]
- Divide by 4:
[tex]\[ x < 2 \][/tex]
### Inequality 2: [tex]\(-2x + 5 \leq -10\)[/tex]
1. Isolate the variable [tex]\(x\)[/tex]:
- Subtract 5 from both sides:
[tex]\[ -2x \leq -15 \][/tex]
2. Divide by -2 and reverse the inequality sign (since we are dividing by a negative number):
[tex]\[ x \geq 7.5 \][/tex]
### Combining the Solutions
The solution for the compound inequality is the union of the two individual solutions, which can be written as:
[tex]\[ x < 2 \quad \text{or} \quad x \geq 7.5 \][/tex]
### Graphing the Combined Inequalities
1. Draw a number line.
2. Plot the solutions:
- For [tex]\(x < 2\)[/tex], shade the region to the left of 2 and draw an open circle at 2 to indicate that 2 is not included.
- For [tex]\(x \geq 7.5\)[/tex], shade the region to the right of 7.5 and draw a closed circle at 7.5 to indicate that 7.5 is included.
The graph of the inequality [tex]\(2(2x - 1) + 7 < 13\)[/tex] or [tex]\(-2x + 5 \leq -10\)[/tex] will look like this:
```
<---(====O----------------]====>
x < 2 x ≥ 7.5
```
Where:
- O indicates an open circle at [tex]\(x = 2\)[/tex] (not included).
- ] indicates a closed circle at [tex]\(x = 7.5\)[/tex] (included).
- The shading on the left represents all [tex]\(x < 2\)[/tex].
- The shading on the right represents all [tex]\(x \geq 7.5\)[/tex].
Thus, the solution set for the given compound inequality is all [tex]\(x\)[/tex] such that [tex]\(x < 2\)[/tex] or [tex]\(x \geq 7.5\)[/tex].
### Inequality 1: [tex]\(2(2x - 1) + 7 < 13\)[/tex]
1. Distribute the 2 on the left side:
[tex]\[ 2(2x - 1) + 7 < 13 \][/tex]
[tex]\[ 4x - 2 + 7 < 13 \][/tex]
2. Combine like terms:
[tex]\[ 4x + 5 < 13 \][/tex]
3. Isolate the variable [tex]\(x\)[/tex]:
- Subtract 5 from both sides:
[tex]\[ 4x < 8 \][/tex]
- Divide by 4:
[tex]\[ x < 2 \][/tex]
### Inequality 2: [tex]\(-2x + 5 \leq -10\)[/tex]
1. Isolate the variable [tex]\(x\)[/tex]:
- Subtract 5 from both sides:
[tex]\[ -2x \leq -15 \][/tex]
2. Divide by -2 and reverse the inequality sign (since we are dividing by a negative number):
[tex]\[ x \geq 7.5 \][/tex]
### Combining the Solutions
The solution for the compound inequality is the union of the two individual solutions, which can be written as:
[tex]\[ x < 2 \quad \text{or} \quad x \geq 7.5 \][/tex]
### Graphing the Combined Inequalities
1. Draw a number line.
2. Plot the solutions:
- For [tex]\(x < 2\)[/tex], shade the region to the left of 2 and draw an open circle at 2 to indicate that 2 is not included.
- For [tex]\(x \geq 7.5\)[/tex], shade the region to the right of 7.5 and draw a closed circle at 7.5 to indicate that 7.5 is included.
The graph of the inequality [tex]\(2(2x - 1) + 7 < 13\)[/tex] or [tex]\(-2x + 5 \leq -10\)[/tex] will look like this:
```
<---(====O----------------]====>
x < 2 x ≥ 7.5
```
Where:
- O indicates an open circle at [tex]\(x = 2\)[/tex] (not included).
- ] indicates a closed circle at [tex]\(x = 7.5\)[/tex] (included).
- The shading on the left represents all [tex]\(x < 2\)[/tex].
- The shading on the right represents all [tex]\(x \geq 7.5\)[/tex].
Thus, the solution set for the given compound inequality is all [tex]\(x\)[/tex] such that [tex]\(x < 2\)[/tex] or [tex]\(x \geq 7.5\)[/tex].
