Sure, let's solve the given inequality step by step:
The given inequality is:
[tex]\[ -4(x + 7) < 3(x - 2) \][/tex]
1. Distribute the numbers on both sides:
Start by distributing [tex]\(-4\)[/tex] and [tex]\(3\)[/tex] on the respective sides of the inequality.
[tex]\[ -4(x + 7) = -4 \cdot x + (-4) \cdot 7 = -4x - 28 \][/tex]
[tex]\[ 3(x - 2) = 3 \cdot x + 3 \cdot (-2) = 3x - 6 \][/tex]
So, the inequality now is:
[tex]\[ -4x - 28 < 3x - 6 \][/tex]
2. Combine like terms by moving all terms involving [tex]\(x\)[/tex] to one side:
To solve for [tex]\(x\)[/tex], we need to move the [tex]\(x\)[/tex]-terms on one side and the constant terms on the other side.
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[ -4x - 3x - 28 < -6 \][/tex]
Simplify:
[tex]\[ -7x - 28 < -6 \][/tex]
3. Isolate the [tex]\(x\)[/tex]-term:
Add [tex]\(28\)[/tex] to both sides to isolate the [tex]\(x\)[/tex]-term.
[tex]\[ -7x - 28 + 28 < -6 + 28 \][/tex]
Simplify:
[tex]\[ -7x < 22 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], we need to divide both sides by [tex]\(-7\)[/tex]. Remember, when you divide by a negative number, the inequality sign reverses direction.
[tex]\[ x > \frac{22}{-7} \][/tex]
Simplify:
[tex]\[ x < -\frac{22}{7} \][/tex]
Therefore, the inequality that is equivalent to [tex]\( -4(x + 7) < 3(x - 2) \)[/tex] is:
[tex]\[ -7x < 22 \][/tex]
Hence, the correct answer is:
D. [tex]\( -7x < 22 \)[/tex]
