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To determine which statement is true given the inequalities [tex]\(5(4-x) < y + 12\)[/tex] and [tex]\(y + 12 < 3x + 1\)[/tex], let's analyze each of the provided options step-by-step:
1. Option 1: [tex]\(3x + 1 - 5(4 - x) = y + 12\)[/tex]
Expanding and simplifying:
[tex]\[ 3x + 1 - 5(4 - x) = y + 12 \][/tex]
[tex]\[ 3x + 1 - 20 + 5x = y + 12 \][/tex]
[tex]\[ 8x - 19 = y + 12 \][/tex]
Substituting [tex]\(y + 12\)[/tex] does not yield both inequalities simultaneously, so we will consider other options.
2. Option 2: [tex]\(5(4 - x) + 3x + 1 = y + 12\)[/tex]
Expanding and simplifying:
[tex]\[ 5(4 - x) + 3x + 1 = y + 12 \][/tex]
[tex]\[ 20 - 5x + 3x + 1 = y + 12 \][/tex]
[tex]\[ 20 - 2x + 1 = y + 12 \][/tex]
[tex]\[ 21 - 2x = y + 12 \][/tex]
This equation does not fit the form of the given inequalities simultaneously, which leads us to check other options.
3. Option 3: [tex]\(3x + 1 < 5(4 + x)\)[/tex]
Expanding and simplifying:
[tex]\[ 3x + 1 < 5(4 + x) \][/tex]
[tex]\[ 3x + 1 < 20 + 5x \][/tex]
[tex]\[ 3x + 1 - 5x < 20 \][/tex]
[tex]\[ -2x + 1 < 20 \][/tex]
[tex]\[ -2x < 19 \][/tex]
[tex]\[ x > -\frac{19}{2} \][/tex]
This simplifies to [tex]\(x > -\frac{19}{2}\)[/tex]. However, this inequality being true does not guarantee that the given conditions hold simultaneously.
4. Option 4: [tex]\(5(4 - x) < 3x + 1\)[/tex]
Expanding and simplifying:
[tex]\[ 5(4 - x) < 3x + 1 \][/tex]
[tex]\[ 20 - 5x < 3x + 1 \][/tex]
[tex]\[ 20 - 1 < 3x + 5x \][/tex]
[tex]\[ 19 < 8x \][/tex]
[tex]\[ x > \frac{19}{8} \][/tex]
This indicates that [tex]\(x > \frac{19}{8}\)[/tex].
When [tex]\(x > \frac{19}{8}\)[/tex], it is consistent with the given inequalities [tex]\(5(4 - x) < y + 12\)[/tex] and [tex]\(y + 12 < 3x + 1\)[/tex]. Therefore, this inequality ensures that both conditions are satisfied.
Hence, the correct statement is:
[tex]\[ \boxed{5(4 - x) < 3x + 1} \][/tex]
1. Option 1: [tex]\(3x + 1 - 5(4 - x) = y + 12\)[/tex]
Expanding and simplifying:
[tex]\[ 3x + 1 - 5(4 - x) = y + 12 \][/tex]
[tex]\[ 3x + 1 - 20 + 5x = y + 12 \][/tex]
[tex]\[ 8x - 19 = y + 12 \][/tex]
Substituting [tex]\(y + 12\)[/tex] does not yield both inequalities simultaneously, so we will consider other options.
2. Option 2: [tex]\(5(4 - x) + 3x + 1 = y + 12\)[/tex]
Expanding and simplifying:
[tex]\[ 5(4 - x) + 3x + 1 = y + 12 \][/tex]
[tex]\[ 20 - 5x + 3x + 1 = y + 12 \][/tex]
[tex]\[ 20 - 2x + 1 = y + 12 \][/tex]
[tex]\[ 21 - 2x = y + 12 \][/tex]
This equation does not fit the form of the given inequalities simultaneously, which leads us to check other options.
3. Option 3: [tex]\(3x + 1 < 5(4 + x)\)[/tex]
Expanding and simplifying:
[tex]\[ 3x + 1 < 5(4 + x) \][/tex]
[tex]\[ 3x + 1 < 20 + 5x \][/tex]
[tex]\[ 3x + 1 - 5x < 20 \][/tex]
[tex]\[ -2x + 1 < 20 \][/tex]
[tex]\[ -2x < 19 \][/tex]
[tex]\[ x > -\frac{19}{2} \][/tex]
This simplifies to [tex]\(x > -\frac{19}{2}\)[/tex]. However, this inequality being true does not guarantee that the given conditions hold simultaneously.
4. Option 4: [tex]\(5(4 - x) < 3x + 1\)[/tex]
Expanding and simplifying:
[tex]\[ 5(4 - x) < 3x + 1 \][/tex]
[tex]\[ 20 - 5x < 3x + 1 \][/tex]
[tex]\[ 20 - 1 < 3x + 5x \][/tex]
[tex]\[ 19 < 8x \][/tex]
[tex]\[ x > \frac{19}{8} \][/tex]
This indicates that [tex]\(x > \frac{19}{8}\)[/tex].
When [tex]\(x > \frac{19}{8}\)[/tex], it is consistent with the given inequalities [tex]\(5(4 - x) < y + 12\)[/tex] and [tex]\(y + 12 < 3x + 1\)[/tex]. Therefore, this inequality ensures that both conditions are satisfied.
Hence, the correct statement is:
[tex]\[ \boxed{5(4 - x) < 3x + 1} \][/tex]
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