To solve the given inequality:
[tex]\[
-4(x+7) < 3(x-2)
\][/tex]
we will proceed step-by-step to derive the equivalent inequality.
1. Distribute the constants within the parentheses:
[tex]\[
-4(x + 7) < 3(x - 2)
\][/tex]
[tex]\[
-4x - 28 < 3x - 6
\][/tex]
2. Combine like terms by adding [tex]\(4x\)[/tex] to both sides:
[tex]\[
-28 < 3x - 6 + 4x
\][/tex]
[tex]\[
-28 < 7x - 6
\][/tex]
3. Add 6 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
-28 + 6 < 7x
\][/tex]
[tex]\[
-22 < 7x
\][/tex]
4. Divide both sides by 7 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{-22}{7} < x
\][/tex]
[tex]\[
x > \frac{-22}{7}
\][/tex]
5. To express the result in a form that matches the given multiple-choice answers, we can reverse the inequality and terms, yielding:
[tex]\[
7x > -22
\][/tex]
or in other terms:
[tex]\[
-7x < 22
\][/tex]
Therefore, the equivalent inequality is:
[tex]\[
\boxed{-7x < 22}
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
