To find an inequality equivalent to the given one, we start with the original inequality and simplify.
Original inequality:
[tex]\[
-4(x + 7) < 3(x - 2)
\][/tex]
Step 1: Distribute the constants within the parentheses.
Left side:
[tex]\[
-4(x + 7) = -4x - 28
\][/tex]
Right side:
[tex]\[
3(x - 2) = 3x - 6
\][/tex]
So the inequality becomes:
[tex]\[
-4x - 28 < 3x - 6
\][/tex]
Step 2: Combine like terms. First, let's move all terms involving [tex]\(x\)[/tex] to one side and constants to the other.
Add [tex]\(4x\)[/tex] to both sides:
[tex]\[
-28 < 7x - 6
\][/tex]
Step 3: Isolate the variable [tex]\(x\)[/tex]. Add 6 to both sides:
[tex]\[
-28 + 6 < 7x
\][/tex]
[tex]\[
-22 < 7x
\][/tex]
Step 4: Rewrite the inequality to isolate [tex]\(x\)[/tex]. Divide both sides by 7 (keeping in mind that dividing both sides of an inequality by a positive number does not change the direction of the inequality):
[tex]\[
-\frac{22}{7} < x
\][/tex]
This can also be written as:
[tex]\[
x > -\frac{22}{7}
\][/tex]
Step 5: Rewrite the inequality in standard form for comparison if needed. Inverting it results in:
[tex]\[
7x > -22
\][/tex]
But we should normalize it back as the original format:
[tex]\[
-7x < 22
\][/tex]
So, the equivalent inequality to [tex]\( -4(x + 7) < 3(x - 2) \)[/tex] is:
[tex]\[
-7x < 22
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{-7x < 22}
\][/tex]
