Sure, let's solve the inequality step-by-step and determine which of the given values for [tex]\( x \)[/tex] satisfy it.
The original inequality is:
[tex]\[ 4(2 - x) > -2x - 3(4x + 1) \][/tex]
First, we distribute the terms inside the parentheses:
[tex]\[
4 \cdot 2 - 4 \cdot x > -2x - 3(4x + 1) \\
8 - 4x > -2x - 12x - 3
\][/tex]
Next, simplify and combine like terms:
[tex]\[
8 - 4x > -14x - 3
\][/tex]
To isolate [tex]\( x \)[/tex], we add [tex]\( 14x \)[/tex] to both sides:
[tex]\[
8 - 4x + 14x > -14x + 14x - 3 \\
8 + 10x > -3
\][/tex]
Now, we add 3 to both sides:
[tex]\[
8 + 3 + 10x > -3 + 3 \\
11 + 10x > 0
\][/tex]
Finally, solve for [tex]\( x \)[/tex]:
[tex]\[
11 + 10x > 0 \\
10x > -11 \\
x > -1.1
\][/tex]
Thus, the inequality [tex]\( 4(2 - x) > -2x - 3(4x + 1) \)[/tex] simplifies to [tex]\( x > -1.1 \)[/tex].
Now, let’s determine which of the given [tex]\( x \)[/tex]-values satisfy this inequality:
1. [tex]\( x = -1.1 \)[/tex]:
[tex]\[ -1.1 \not> -1.1 \][/tex] (False)
2. [tex]\( x = -2.2 \)[/tex]:
[tex]\[ -2.2 \not> -1.1 \][/tex] (False)
3. [tex]\( x = 0 \)[/tex]:
[tex]\[ 0 > -1.1 \][/tex] (True)
4. [tex]\( x = -10 \)[/tex]:
[tex]\[ -10 \not> -1.1 \][/tex] (False)
5. [tex]\( x = 10 \)[/tex]:
[tex]\[ 10 > -1.1 \][/tex] (True)
The values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( 4(2 - x) > -2x - 3(4x + 1) \)[/tex] are:
[tex]\[ x = 0 \quad \text{and} \quad x = 10 \][/tex]
