Find the best solutions to your problems with the help of IDNLearn.com's experts. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.

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

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

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


Sagot :

To solve the inequality [tex]\(-4(x+7) < 3(x-2)\)[/tex] step-by-step, we need to simplify both sides of the inequality and isolate [tex]\(x\)[/tex].

1. Expand the expressions:
[tex]\[ -4(x + 7) < 3(x - 2) \][/tex]

Expanding both sides:
[tex]\[ -4x - 28 < 3x - 6 \][/tex]

2. Isolate [tex]\(x\)[/tex] on one side of the inequality:
First, add [tex]\(4x\)[/tex] to both sides to eliminate [tex]\(-4x\)[/tex] from the left side:
[tex]\[ -28 < 3x - 6 + 4x \][/tex]
Simplify:
[tex]\[ -28 < 7x - 6 \][/tex]

Next, add 6 to both sides to eliminate [tex]\(-6\)[/tex] from the right side:
[tex]\[ -28 + 6 < 7x \][/tex]
Simplify:
[tex]\[ -22 < 7x \][/tex]

This can also be written as:
[tex]\[ 7x > -22 \][/tex]

3. Final inequality:

Dividing both sides by 7 changes the expression slightly:
[tex]\[ x > -\frac{22}{7} \][/tex]

Thus the equivalent inequality to the given [tex]\(-4(x+7) < 3(x-2)\)[/tex] is [tex]\(7x > -22\)[/tex].

4. Match with the given options:

To match the provided answer choices:
[tex]\[ 7x > -22 \][/tex]
can be written multiplicative inverse:
[tex]\[ -7 x < 22 \][/tex]

So, the correct answer is:
[tex]\[ \boxed{-7 x < 22} \][/tex]

Thus the correct option is:
D. [tex]\(-7 x < 22\)[/tex]