Question
(a)
Martin is considering the expressions 12(7x+48)21(7x+48) and −(12x−3)+4(x+5)−(21x−3)+4(x+5). He wants to know if one expression is greater than the other for all values of xx.
Part A
Which statement about the relationship between the expressions is true?
A
The value of the expression 12(7x+48)\frac{1}{2}\left(7x+48\right)21(7x+48) is always equal to the value of the expression −(12x−3)+(x+5)-\left(\frac{1}{2}x-3\right)+\left(x+5\right)−(21x−3)+(x+5).
B
The value of the expression 12(7x+48)\frac{1}{2}\left(7x+48\right)21(7x+48) is always less than the value of the expression −(12x−3)+4(x+5)-\left(\frac{1}{2}x-3\right)+4\left(x+5\right)−(21x−3)+4(x+5).
C
The value of the expression 12(7x+48)\frac{1}{2}\left(7x+48\right)21(7x+48) is always greater than the value of the expression −(12x−3)+4(x+5)-\left(\frac{1}{2}x-3\right)+4\left(x+5\right)−(21x−3)+4(x+5).
D
The value of the expression 12(7x+48)\frac{1}{2}\left(7x+48\right)21(7x+48)is sometimes greater than and sometimes less than the value of the expression −(12x−3)+4(x+5)-\left(\frac{1}{2}x-3\right)+4\left(x+5\right)−(21x−3)+4(x+5).
Answers
GPT-4o mini
To analyze the expressions and determine their relationship, we first need to simplify both expressions provided.
1. **First expression:**
\(\frac{1}{2}(7x + 48)\)
Calculating this gets us:
\[
\frac{1}{2}(7x + 48) = \frac{7}{2}x + 24
\]
2. **Second expression:**
\(-\left(12x - 3\right) + 4(x + 5) - (21x - 3) + 4(x + 5)\)
Simplifying this:
- Start by distributing:
\[
- (12x - 3) = -12x + 3
\]
\[
4(x + 5) = 4x + 20
\]
Thus, combining these parts, we have:
\[
-12x + 3 + 4x + 20 - (21x - 3) + 4x + 20
\]
Now simplify:
- For \(-(21x - 3) = -21x + 3\)
Putting it all together:
\[
-12x + 3 + 4x + 20 - 21x + 3 + 4x + 20
\]
Combine like terms:
- \( -12x + 4x - 21x + 4x = -25x \)
- \( 3 + 20 + 3 + 20 = 46 \)
Thus, the second expression simplifies to:
\[
-25x + 46
\]
Now we have the simplified expressions:
1. First expression: \(\frac{7}{2}x + 24\)
2. Second expression: \(-25x + 46\)
Next, we can compare these two expressions:
1. Set up the inequality:
\[
\frac{7}{2}x + 24 > -25x + 46
\]
2. Combine the terms:
\[
\frac{7}{2}x + 25x > 46 - 24
\]
This combines to:
\[
\frac{7}{2}x + \frac{50}{2}x > 22
\]
Which gives:
\[
\frac{57}{2}x > 22
\]
Dividing both sides by \(\frac{57}{2}\):
\[
x > \frac{22 \cdot 2}{57}
\]
\[
x > \frac{44}{57}
\]
This shows that \(\frac{7}{2}x + 24\) is greater than \(-25x + 46\) only for \(x > \frac{44}{57}\) and not for all \(x\).
Thus, the correct statement regarding the relationship between the expressions is:
**D**: The value of the expression \( \frac{1}{2}(7x + 48) \) is sometimes greater than and sometimes less than the value of the expression \( - (12x - 3) + 4(x + 5) - (21x - 3) + 4(x + 5) \).
1. **First expression:**
\(\frac{1}{2}(7x + 48)\)
Calculating this gets us:
\[
\frac{1}{2}(7x + 48) = \frac{7}{2}x + 24
\]
2. **Second expression:**
\(-\left(12x - 3\right) + 4(x + 5) - (21x - 3) + 4(x + 5)\)
Simplifying this:
- Start by distributing:
\[
- (12x - 3) = -12x + 3
\]
\[
4(x + 5) = 4x + 20
\]
Thus, combining these parts, we have:
\[
-12x + 3 + 4x + 20 - (21x - 3) + 4x + 20
\]
Now simplify:
- For \(-(21x - 3) = -21x + 3\)
Putting it all together:
\[
-12x + 3 + 4x + 20 - 21x + 3 + 4x + 20
\]
Combine like terms:
- \( -12x + 4x - 21x + 4x = -25x \)
- \( 3 + 20 + 3 + 20 = 46 \)
Thus, the second expression simplifies to:
\[
-25x + 46
\]
Now we have the simplified expressions:
1. First expression: \(\frac{7}{2}x + 24\)
2. Second expression: \(-25x + 46\)
Next, we can compare these two expressions:
1. Set up the inequality:
\[
\frac{7}{2}x + 24 > -25x + 46
\]
2. Combine the terms:
\[
\frac{7}{2}x + 25x > 46 - 24
\]
This combines to:
\[
\frac{7}{2}x + \frac{50}{2}x > 22
\]
Which gives:
\[
\frac{57}{2}x > 22
\]
Dividing both sides by \(\frac{57}{2}\):
\[
x > \frac{22 \cdot 2}{57}
\]
\[
x > \frac{44}{57}
\]
This shows that \(\frac{7}{2}x + 24\) is greater than \(-25x + 46\) only for \(x > \frac{44}{57}\) and not for all \(x\).
Thus, the correct statement regarding the relationship between the expressions is:
**D**: The value of the expression \( \frac{1}{2}(7x + 48) \) is sometimes greater than and sometimes less than the value of the expression \( - (12x - 3) + 4(x + 5) - (21x - 3) + 4(x + 5) \).