Question
(a)
Martin is considering the expressions
and
. He wants to know if one expression is greater than the other for all values of
.
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)
2
1
(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)−(
2
1
x−3)+(x+5).
B
The value of the expression 12(7x+48)\frac{1}{2}\left(7x+48\right)
2
1
(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)−(
2
1
x−3)+4(x+5).
C
The value of the expression 12(7x+48)\frac{1}{2}\left(7x+48\right)
2
1
(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)−(
2
1
x−3)+4(x+5).
D
The value of the expression 12(7x+48)\frac{1}{2}\left(7x+48\right)
2
1
(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)−(
2
1
x−3)+4(x+5).
(b)
Part B
Show or explain how you found your answer to Part A.
Enter your work or your explanation in the box provided.
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(c)
Part C
Write a new expression that always has a greater value than both of these expressions.
Enter your expressions in the box provided.
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Answers
GPT-4o mini
To analyze the expressions provided, let's first clarify and evaluate them:
1. **Expression A**:
\[
\frac{1}{2}(7x + 48) = \frac{7}{2}x + 24
\]
2. **Expression B**:
\[
-(12x - 3) + (x + 5) - \left(\frac{1}{2}x - 3\right) + (x + 5)
\]
This simplifies as follows:
\[
-12x + 3 + x + 5 - \frac{1}{2}x + 3 + x + 5
\]
Combine the terms:
\[
-12x + x - \frac{1}{2}x + 3 + 5 + 3 + 5 = -12x + 2x - \frac{1}{2}x + 16 = -\frac{24}{2}x + \frac{4}{2}x + 16 = -\frac{20}{2}x + 16 = -10x + 16
\]
3. **Expression C**:
\[
-(12x - 3) + 4(x + 5) - \left(\frac{1}{2}x - 3\right) + 4(x + 5)
\]
This simplifies as follows:
\[
-12x + 3 + 4x + 20 - \frac{1}{2}x + 3 + 4x + 20
\]
Combine the terms:
\[
-12x + 4x + 4x - \frac{1}{2}x + 3 + 20 + 3 + 20 = (-12x + 8x - \frac{1}{2}x) + 46
\]
This can be rewritten as:
\[
-\left(12 - 8 + \frac{1}{2}\right)x + 46 = -\frac{24}{2}x + \frac{16}{2}x + 46 = -\frac{8}{2}x + 46 = -10x + 46
\]
Next, we compare the original expressions:
- **Expression A**:
\[
\frac{7}{2}x + 24
\]
- **Expression B**:
\[
-10x + 16
\]
- **Expression C**:
\[
-10x + 46
\]
Now, comparing:
- For **Part A**, we need to determine the relationship between:
\[
\frac{7}{2}x + 24 \quad \text{(Expression A)} \quad \text{and} \quad -10x + 46 \quad \text{(Expression C)}
\]
To find this, equate the two:
\[
\frac{7}{2}x + 24 = -10x + 46
\]
Solving for \( x \):
\[
\frac{7}{2}x + 10x = 46 - 24
\]
\[
\frac{7}{2}x + \frac{20}{2}x = 22
\]
\[
\frac{27}{2}x = 22
\]
\[
x = \frac{22 \times 2}{27} = \frac{44}{27}
\]
Thus, we see that the expressions could intersect, showing **D**: “The value of the expression \(\frac{1}{2}(7x + 48)\) is sometimes greater than and sometimes less than the value of the expression \(-10x + 46\)” is indeed correct.
**Part B Explanation**:
To find the relationship between the expressions, we simplified both and compared them. We determined the point of intersection, which tells us that both expressions don't consistently hold a singular magnitude relationship across all \(x\).
**Part C**:
To create an expression greater than both, observe that since \( -10x + 46 \) is always being held against \( \frac{7}{2}x + 24 \), one possible expression that could always be greater could simply be:
\[
-10x + 50
\]
This is \(4\) units greater than Expression C, ensuring it is always higher, regardless of \(x\).
Thus, my answer for **Part C** is **\(-10x + 50\)**.
1. **Expression A**:
\[
\frac{1}{2}(7x + 48) = \frac{7}{2}x + 24
\]
2. **Expression B**:
\[
-(12x - 3) + (x + 5) - \left(\frac{1}{2}x - 3\right) + (x + 5)
\]
This simplifies as follows:
\[
-12x + 3 + x + 5 - \frac{1}{2}x + 3 + x + 5
\]
Combine the terms:
\[
-12x + x - \frac{1}{2}x + 3 + 5 + 3 + 5 = -12x + 2x - \frac{1}{2}x + 16 = -\frac{24}{2}x + \frac{4}{2}x + 16 = -\frac{20}{2}x + 16 = -10x + 16
\]
3. **Expression C**:
\[
-(12x - 3) + 4(x + 5) - \left(\frac{1}{2}x - 3\right) + 4(x + 5)
\]
This simplifies as follows:
\[
-12x + 3 + 4x + 20 - \frac{1}{2}x + 3 + 4x + 20
\]
Combine the terms:
\[
-12x + 4x + 4x - \frac{1}{2}x + 3 + 20 + 3 + 20 = (-12x + 8x - \frac{1}{2}x) + 46
\]
This can be rewritten as:
\[
-\left(12 - 8 + \frac{1}{2}\right)x + 46 = -\frac{24}{2}x + \frac{16}{2}x + 46 = -\frac{8}{2}x + 46 = -10x + 46
\]
Next, we compare the original expressions:
- **Expression A**:
\[
\frac{7}{2}x + 24
\]
- **Expression B**:
\[
-10x + 16
\]
- **Expression C**:
\[
-10x + 46
\]
Now, comparing:
- For **Part A**, we need to determine the relationship between:
\[
\frac{7}{2}x + 24 \quad \text{(Expression A)} \quad \text{and} \quad -10x + 46 \quad \text{(Expression C)}
\]
To find this, equate the two:
\[
\frac{7}{2}x + 24 = -10x + 46
\]
Solving for \( x \):
\[
\frac{7}{2}x + 10x = 46 - 24
\]
\[
\frac{7}{2}x + \frac{20}{2}x = 22
\]
\[
\frac{27}{2}x = 22
\]
\[
x = \frac{22 \times 2}{27} = \frac{44}{27}
\]
Thus, we see that the expressions could intersect, showing **D**: “The value of the expression \(\frac{1}{2}(7x + 48)\) is sometimes greater than and sometimes less than the value of the expression \(-10x + 46\)” is indeed correct.
**Part B Explanation**:
To find the relationship between the expressions, we simplified both and compared them. We determined the point of intersection, which tells us that both expressions don't consistently hold a singular magnitude relationship across all \(x\).
**Part C**:
To create an expression greater than both, observe that since \( -10x + 46 \) is always being held against \( \frac{7}{2}x + 24 \), one possible expression that could always be greater could simply be:
\[
-10x + 50
\]
This is \(4\) units greater than Expression C, ensuring it is always higher, regardless of \(x\).
Thus, my answer for **Part C** is **\(-10x + 50\)**.