Select the correct answer.

Which inequality is equivalent to the given inequality?
[tex]\[ -4(x+7)\ \textless \ 3(x-2) \][/tex]

A. [tex]\(-7x \ \textgreater \ -34\)[/tex]

B. [tex]\(-7x \ \textgreater \ 22\)[/tex]

C. [tex]\(-7x \ \textless \ -34\)[/tex]

D. [tex]\(-7x \ \textless \ 22\)[/tex]



Answer :

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]