To solve the inequality [tex]\(-4(x+7) < 3(x-2)\)[/tex] step-by-step, we need to simplify both sides of the inequality and isolate [tex]\(x\)[/tex].
1. Expand the expressions:
[tex]\[
-4(x + 7) < 3(x - 2)
\][/tex]
Expanding both sides:
[tex]\[
-4x - 28 < 3x - 6
\][/tex]
2. Isolate [tex]\(x\)[/tex] on one side of the inequality:
First, add [tex]\(4x\)[/tex] to both sides to eliminate [tex]\(-4x\)[/tex] from the left side:
[tex]\[
-28 < 3x - 6 + 4x
\][/tex]
Simplify:
[tex]\[
-28 < 7x - 6
\][/tex]
Next, add 6 to both sides to eliminate [tex]\(-6\)[/tex] from the right side:
[tex]\[
-28 + 6 < 7x
\][/tex]
Simplify:
[tex]\[
-22 < 7x
\][/tex]
This can also be written as:
[tex]\[
7x > -22
\][/tex]
3. Final inequality:
Dividing both sides by 7 changes the expression slightly:
[tex]\[
x > -\frac{22}{7}
\][/tex]
Thus the equivalent inequality to the given [tex]\(-4(x+7) < 3(x-2)\)[/tex] is [tex]\(7x > -22\)[/tex].
4. Match with the given options:
To match the provided answer choices:
[tex]\[
7x > -22
\][/tex]
can be written multiplicative inverse:
[tex]\[
-7 x < 22
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{-7 x < 22}
\][/tex]
Thus the correct option is:
D. [tex]\(-7 x < 22\)[/tex]
