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Solve the compound inequality. Write the solution set.

[tex]\[ 2x + 6 \ \textgreater \ -11 \text{ and } 2x + 6 \ \textless \ 10 \][/tex]

Sagot :

GinnyAnswer
Let's solve the compound inequality step by step.

We have two inequalities:
1. [tex]\(2x + 6 > -11\)[/tex]
2. [tex]\(2x + 6 < 10\)[/tex]

We need to solve each inequality separately.

### Solving the first inequality:
[tex]\[2x + 6 > -11\][/tex]

Step 1: Subtract 6 from both sides to isolate the term with the variable [tex]\(x\)[/tex]:
[tex]\[ 2x + 6 - 6 > -11 - 6 \][/tex]
[tex]\[ 2x > -17 \][/tex]

Step 2: Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{2x}{2} > \frac{-17}{2} \][/tex]
[tex]\[ x > -\frac{17}{2} \][/tex]

### Solving the second inequality:
[tex]\[2x + 6 < 10\][/tex]

Step 1: Subtract 6 from both sides to isolate the term with the variable [tex]\(x\)[/tex]:
[tex]\[ 2x + 6 - 6 < 10 - 6 \][/tex]
[tex]\[ 2x < 4 \][/tex]

Step 2: Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{2x}{2} < \frac{4}{2} \][/tex]
[tex]\[ x < 2 \][/tex]

### Combining the results:

We now combine the solutions from both inequalities:
[tex]\[ -\frac{17}{2} < x < 2 \][/tex]

This means [tex]\(x\)[/tex] must be greater than [tex]\(-\frac{17}{2}\)[/tex] and less than 2. The solution set can be written in interval notation as:
[tex]\[ \left( -\frac{17}{2}, 2 \right) \][/tex]

### Conclusion:
The solution set for the compound inequality [tex]\(2x + 6 > -11\)[/tex] and [tex]\(2x + 6 < 10\)[/tex] is [tex]\(\left( -\frac{17}{2}, 2 \right)\)[/tex].
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