Answered

At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

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]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.