Answer :
To solve for [tex]\( x \)[/tex] in the compound inequality:
[tex]\[ 5x - 10 > 20 \text{ or } 5x - 10 \leq -15 \][/tex]
we will break it into two separate inequalities and solve for [tex]\( x \)[/tex] in each part.
### Solving the First Inequality:
[tex]\[ 5x - 10 > 20 \][/tex]
1. Add 10 to both sides:
[tex]\[ 5x - 10 + 10 > 20 + 10 \][/tex]
[tex]\[ 5x > 30 \][/tex]
2. Divide both sides by 5:
[tex]\[ \frac{5x}{5} > \frac{30}{5} \][/tex]
[tex]\[ x > 6 \][/tex]
Thus, the solution to the first inequality is:
[tex]\[ x > 6 \][/tex]
### Solving the Second Inequality:
[tex]\[ 5x - 10 \leq -15 \][/tex]
1. Add 10 to both sides:
[tex]\[ 5x - 10 + 10 \leq -15 + 10 \][/tex]
[tex]\[ 5x \leq -5 \][/tex]
2. Divide both sides by 5:
[tex]\[ \frac{5x}{5} \leq \frac{-5}{5} \][/tex]
[tex]\[ x \leq -1 \][/tex]
Thus, the solution to the second inequality is:
[tex]\[ x \leq -1 \][/tex]
### Combining the Results:
We are looking for the union of the solutions to the two inequalities because we have an "or" condition. Therefore, we combine both solution sets:
[tex]\[ x \leq -1 \text{ or } x > 6 \][/tex]
### Conclusion:
Comparing this result to the provided options, the correct solution is:
[tex]\[ x \leq -1 \text{ or } x > 6 \][/tex]
[tex]\[ 5x - 10 > 20 \text{ or } 5x - 10 \leq -15 \][/tex]
we will break it into two separate inequalities and solve for [tex]\( x \)[/tex] in each part.
### Solving the First Inequality:
[tex]\[ 5x - 10 > 20 \][/tex]
1. Add 10 to both sides:
[tex]\[ 5x - 10 + 10 > 20 + 10 \][/tex]
[tex]\[ 5x > 30 \][/tex]
2. Divide both sides by 5:
[tex]\[ \frac{5x}{5} > \frac{30}{5} \][/tex]
[tex]\[ x > 6 \][/tex]
Thus, the solution to the first inequality is:
[tex]\[ x > 6 \][/tex]
### Solving the Second Inequality:
[tex]\[ 5x - 10 \leq -15 \][/tex]
1. Add 10 to both sides:
[tex]\[ 5x - 10 + 10 \leq -15 + 10 \][/tex]
[tex]\[ 5x \leq -5 \][/tex]
2. Divide both sides by 5:
[tex]\[ \frac{5x}{5} \leq \frac{-5}{5} \][/tex]
[tex]\[ x \leq -1 \][/tex]
Thus, the solution to the second inequality is:
[tex]\[ x \leq -1 \][/tex]
### Combining the Results:
We are looking for the union of the solutions to the two inequalities because we have an "or" condition. Therefore, we combine both solution sets:
[tex]\[ x \leq -1 \text{ or } x > 6 \][/tex]
### Conclusion:
Comparing this result to the provided options, the correct solution is:
[tex]\[ x \leq -1 \text{ or } x > 6 \][/tex]
