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Select the correct answer.

If [tex]x+12 \leq 5-y[/tex] and [tex]5-y \leq 2(x-3)[/tex], then which statement is true?

A. [tex]x+12 \leq 2(5-y)[/tex]

B. [tex]x+12 \leq 2 x - 3[/tex]

C. [tex]x+12 \leq 2(x-3)[/tex]

D. [tex]x+12 \leq y-5[/tex]

Sagot :

GinnyAnswer
Let's analyze the given inequalities and select the correct statement through step-by-step simplification and comparison.

Given inequalities:
1. [tex]\(x + 12 \leq 5 - y\)[/tex]
2. [tex]\(5 - y \leq 2(x - 3)\)[/tex]

First, let's derive the possible relationships from these inequalities.

### Inequality 1: [tex]\(x + 12 \leq 5 - y\)[/tex]
Rearrange this inequality to combine [tex]\(x\)[/tex] and [tex]\(y\)[/tex] on one side:
[tex]\[ x + y \leq -7 \][/tex]

### Inequality 2: [tex]\(5 - y \leq 2(x - 3)\)[/tex]
Rewrite the right side:
[tex]\[5 - y \leq 2x - 6\][/tex]

Next, rearrange to isolate [tex]\(y\)[/tex]:
[tex]\[ -y \leq 2x - 11 \][/tex]
[tex]\[ y \geq 11 - 2x \][/tex]

Thus, the inequalities we have are:
1. [tex]\(x + y \leq -7\)[/tex]
2. [tex]\(y \geq 11 - 2x\)[/tex]

Now, let's test each statement to find which one matches these inequalities:

### Option A: [tex]\(x + 12 \leq 2(5 - y)\)[/tex]

Simplify the right side:
[tex]\[ x + 12 \leq 10 - 2y \][/tex]

Rearrange to combine [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x + 2y \leq -2 \][/tex]

This result does not necessarily align with [tex]\(x + y \leq -7\)[/tex]. Therefore, option A is not correct.

### Option B: [tex]\(x + 12 \leq 2x - 3\)[/tex]

Rearrange to solve for [tex]\(x\)[/tex]:
[tex]\[ x + 12 \leq 2x - 3 \][/tex]
[tex]\[ 12 \leq x - 3 \][/tex]
[tex]\[ x \geq 15 \][/tex]

While this could be true, it doesn't necessarily depend on the inequalities [tex]\(x + y \leq -7\)[/tex] and [tex]\(y \geq 11 - 2x\)[/tex]. Therefore, option B is not correct.

### Option C: [tex]\(x + 12 \leq 2(x - 3)\)[/tex]

Expand and simplify the right side:
[tex]\[ x + 12 \leq 2x - 6 \][/tex]
[tex]\[ 18 \leq x \][/tex]

This is a reasonable conclusion from the given inequalities and can potentially align with [tex]\(x + y \leq -7\)[/tex]. This appears to be the most plausible statement.

### Option D: [tex]\(x + 12 \leq y - 5\)[/tex]

Rearrange to solve for [tex]\(y\)[/tex]:
[tex]\[ x + 17 \leq y \][/tex]
[tex]\[ y \geq x + 17 \][/tex]

This result does not necessarily match [tex]\(x + y \leq -7\)[/tex] or [tex]\(y \geq 11 - 2x\)[/tex]. Therefore, option D is not correct.

Considering all the analyzed statements, the most plausible and correct statement that aligns with the derived inequalities is:

[tex]\[ C. \, x + 12 \leq 2(x - 3) \][/tex]

So, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
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