To solve the inequality [tex]\(-4(x + 7) < 3(x - 2)\)[/tex], we need to follow a series of algebraic steps. Here's the detailed, step-by-step solution:
1. Apply the distributive property to both sides of the inequality:
[tex]\[-4(x + 7) < 3(x - 2)\][/tex]
Expanding both sides:
[tex]\[
-4 \cdot x - 4 \cdot 7 < 3 \cdot x - 3 \cdot 2
\][/tex]
Simplifying further:
[tex]\[
-4x - 28 < 3x - 6
\][/tex]
2. Isolate the variable [tex]\(x\)[/tex] on one side of the inequality:
To do this, we need to get all the terms containing [tex]\(x\)[/tex] on one side of the inequality and the constant terms on the other side. Start by subtracting [tex]\(3x\)[/tex] from both sides:
[tex]\[
-4x - 28 - 3x < 3x - 6 - 3x
\][/tex]
Simplifying this gives:
[tex]\[
-7x - 28 < -6
\][/tex]
3. Remove the constant term from the left side:
To isolate the term with [tex]\(x\)[/tex], we add 28 to both sides of the inequality:
[tex]\[
-7x - 28 + 28 < -6 + 28
\][/tex]
Simplifying this gives:
[tex]\[
-7x < 22
\][/tex]
4. Determine the correct form of the inequality:
Looking at the simplified inequality, we see that it matches option A among the provided choices:
[tex]\[
-7x < 22
\][/tex]
Therefore, the correct answer is:
A. [tex]\(-7x < 22\)[/tex]
