Answer :
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]
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]
