Answer :
To solve the given compound inequality, we need to deal with both inequalities separately and then combine the results.
1. First inequality: [tex]\( 3x + 2 > 5 \)[/tex]
- Subtract 2 from both sides:
[tex]\[ 3x + 2 - 2 > 5 - 2 \implies 3x > 3 \][/tex]
- Divide both sides by 3:
[tex]\[ \frac{3x}{3} > \frac{3}{3} \implies x > 1 \][/tex]
2. Second inequality: [tex]\( 3x \leq 9 \)[/tex]
- Divide both sides by 3:
[tex]\[ \frac{3x}{3} \leq \frac{9}{3} \implies x \leq 3 \][/tex]
3. Combining the inequalities:
[tex]\[ 1 < x \leq 3 \][/tex]
This means that [tex]\( x \)[/tex] must be greater than 1 but less than or equal to 3.
4. Graphing this inequality on a number line:
- An open circle on 1, indicating that 1 is not included in the solution set.
- A closed circle on 3, indicating that 3 is included in the solution set.
- Shading in between, covering all values between 1 and 3.
So, the correct description of the graph of the compound inequality is:
A number line with an open circle on 1, a closed circle on 3, and shading in between.
1. First inequality: [tex]\( 3x + 2 > 5 \)[/tex]
- Subtract 2 from both sides:
[tex]\[ 3x + 2 - 2 > 5 - 2 \implies 3x > 3 \][/tex]
- Divide both sides by 3:
[tex]\[ \frac{3x}{3} > \frac{3}{3} \implies x > 1 \][/tex]
2. Second inequality: [tex]\( 3x \leq 9 \)[/tex]
- Divide both sides by 3:
[tex]\[ \frac{3x}{3} \leq \frac{9}{3} \implies x \leq 3 \][/tex]
3. Combining the inequalities:
[tex]\[ 1 < x \leq 3 \][/tex]
This means that [tex]\( x \)[/tex] must be greater than 1 but less than or equal to 3.
4. Graphing this inequality on a number line:
- An open circle on 1, indicating that 1 is not included in the solution set.
- A closed circle on 3, indicating that 3 is included in the solution set.
- Shading in between, covering all values between 1 and 3.
So, the correct description of the graph of the compound inequality is:
A number line with an open circle on 1, a closed circle on 3, and shading in between.
