To find the value of [tex]\( x \)[/tex] that solves the inequality [tex]\( -5x - 15 > 10 + 20x \)[/tex], we will follow these steps:
1. Move all terms involving [tex]\( x \)[/tex] to one side of the inequality and constants to the other:
[tex]\[
-5x - 15 - 20x > 10
\][/tex]
2. Combine like terms:
[tex]\[
-25x - 15 > 10
\][/tex]
3. Isolate [tex]\( x \)[/tex] by first moving the constant term to the other side:
[tex]\[
-25x > 10 + 15
\][/tex]
4. Simplify the right-hand side:
[tex]\[
-25x > 25
\][/tex]
5. Divide both sides of the inequality by -25, and remember to reverse the inequality sign because we are dividing by a negative number:
[tex]\[
x < \frac{25}{-25}
\][/tex]
[tex]\[
x < -1
\][/tex]
Now, we need to check the given options:
1. [tex]\( x = -2 \)[/tex]:
[tex]\[
-5(-2) - 15 > 10 + 20(-2)
\][/tex]
Simplifying,
[tex]\[
10 - 15 > 10 - 40
\][/tex]
[tex]\[
-5 > -30
\][/tex]
This is true. So, [tex]\( x = -2 \)[/tex] is a solution.
2. [tex]\( x = -1 \)[/tex]:
[tex]\[
-5(-1) - 15 > 10 + 20(-1)
\][/tex]
Simplifying,
[tex]\[
5 - 15 > 10 - 20
\][/tex]
[tex]\[
-10 > -10
\][/tex]
This is false. So [tex]\( x = -1 \)[/tex] is not a solution.
3. [tex]\( x = 0 \)[/tex]:
[tex]\[
-5(0) - 15 > 10 + 20(0)
\][/tex]
Simplifying,
[tex]\[
-15 > 10
\][/tex]
This is false. So [tex]\( x = 0 \)[/tex] is not a solution.
4. [tex]\( x = 1 \)[/tex]:
[tex]\[
-5(1) - 15 > 10 + 20(1)
\][/tex]
Simplifying,
[tex]\[
-5 - 15 > 10 + 20
\][/tex]
[tex]\[
-20 > 30
\][/tex]
This is false. So [tex]\( x = 1 \)[/tex] is not a solution.
Thus, the value of [tex]\( x \)[/tex] which satisfies the given inequality is [tex]\( \boxed{-2} \)[/tex].
