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Select the correct answer.

Which inequality is equivalent to the given inequality?

[tex]-4(x+7)\ \textless \ 3(x-2)[/tex]

A. [tex]-7x\ \textless \ 22[/tex]

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

C. [tex]-7x\ \textgreater \ -34[/tex]

D. [tex]-7x\ \textless \ -34[/tex]


Sagot :

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]