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.
Part C
Write a new expression that always has a greater value than both of these expressions.
Enter your expressions in the box provided.
Answers
GPT-4o mini
To tackle the problem, we need to break it down step by step, evaluating each expression given and determining if one is greater than the other for all values of \( x \).
### Part A
We need to assess the following expressions:
1. \( \frac{1}{2}(7x + 48) \)
2. \( - (12x - 3) + 4(x + 5) - \left(\frac{1}{2}x - 3\right) + 4(x + 5) \)
First, let's simplify each expression.
**Simplifying Expression 1:**
\[
E_1 = \frac{1}{2}(7x + 48) = \frac{7x}{2} + 24
\]
**Simplifying Expression 2:**
We'll simplify:
\[
E_2 = - (12x - 3) + 4(x + 5) - \left(\frac{1}{2}x - 3\right) + 4(x + 5)
\]
Breaking down each part:
1. \( - (12x - 3) = -12x + 3 \)
2. \( 4(x + 5) = 4x + 20 \) [twice, so it's \( 2(4x + 20) = 8x + 40 \)]
3. \( -\left(\frac{1}{2}x - 3\right) = -\frac{1}{2}x + 3 \)
Combining these:
\[
E_2 = -12x + 3 + 8x + 40 - \frac{1}{2}x + 3 + 8x + 40
\]
Combining like terms:
- \( -12x + 8x - \frac{1}{2}x + 8x = -12x + 16x - \frac{1}{2}x = 4x - \frac{1}{2}x = \frac{8x}{2} - \frac{1}{2}x = \frac{7.5x}{2} = \frac{15x}{4} \)
- Collecting constants: \( 3 + 40 + 3 + 40 = 86 \)
So,
\[
E_2 = \frac{15x}{4} + 86
\]
### Comparing the Expressions
We now compare the two simplified expressions:
\[
E_1 = \frac{7x}{2} + 24
\]
\[
E_2 = \frac{15x}{4} + 86
\]
To compare \( E_1 \) and \( E_2 \), we look for values of \( x \):
\[
\frac{7x}{2} + 24 \geq \frac{15x}{4} + 86
\]
Multiply through by 4 to eliminate fractions:
\[
14x + 96 \geq 15x + 344
\]
Rearranging gives:
\[
96 - 344 \geq 15x - 14x
\]
\[
-248 \geq x
\]
So, for \( x \leq -248 \), \( E_1 \) is at least equal to \( E_2 \), and possibly greater.
For \( x > -248 \), \( E_2 \) becomes greater.
Thus, the correct statement 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 \( - (12x - 3) + 4(x + 5) - \left(\frac{1}{2}x - 3\right) + 4(x + 5) \).
### Part B
**Explanation & Summary**:
We found each expression’s simplified form and compared them. The relationship changes depending on the value of \( x \). This indicates that one expression does not consistently dominate the other; rather, it varies based on the specified range of \( x \).
### Part C
**New Expression**:
To construct a new expression that is always greater than both:
We can determine a constant value greater than our evaluated constants from \( E_1 \) and \( E_2 \).
Choose a simple constant, e.g.,
\[
E_3 = 100
\]
This expression will always be greater than either of the expressions since both \( 24 < 100 \) and \( 86 < 100 \) are true regardless of \( x \).
Thus, one such valid expression is \( 100 \).
### Part A
We need to assess the following expressions:
1. \( \frac{1}{2}(7x + 48) \)
2. \( - (12x - 3) + 4(x + 5) - \left(\frac{1}{2}x - 3\right) + 4(x + 5) \)
First, let's simplify each expression.
**Simplifying Expression 1:**
\[
E_1 = \frac{1}{2}(7x + 48) = \frac{7x}{2} + 24
\]
**Simplifying Expression 2:**
We'll simplify:
\[
E_2 = - (12x - 3) + 4(x + 5) - \left(\frac{1}{2}x - 3\right) + 4(x + 5)
\]
Breaking down each part:
1. \( - (12x - 3) = -12x + 3 \)
2. \( 4(x + 5) = 4x + 20 \) [twice, so it's \( 2(4x + 20) = 8x + 40 \)]
3. \( -\left(\frac{1}{2}x - 3\right) = -\frac{1}{2}x + 3 \)
Combining these:
\[
E_2 = -12x + 3 + 8x + 40 - \frac{1}{2}x + 3 + 8x + 40
\]
Combining like terms:
- \( -12x + 8x - \frac{1}{2}x + 8x = -12x + 16x - \frac{1}{2}x = 4x - \frac{1}{2}x = \frac{8x}{2} - \frac{1}{2}x = \frac{7.5x}{2} = \frac{15x}{4} \)
- Collecting constants: \( 3 + 40 + 3 + 40 = 86 \)
So,
\[
E_2 = \frac{15x}{4} + 86
\]
### Comparing the Expressions
We now compare the two simplified expressions:
\[
E_1 = \frac{7x}{2} + 24
\]
\[
E_2 = \frac{15x}{4} + 86
\]
To compare \( E_1 \) and \( E_2 \), we look for values of \( x \):
\[
\frac{7x}{2} + 24 \geq \frac{15x}{4} + 86
\]
Multiply through by 4 to eliminate fractions:
\[
14x + 96 \geq 15x + 344
\]
Rearranging gives:
\[
96 - 344 \geq 15x - 14x
\]
\[
-248 \geq x
\]
So, for \( x \leq -248 \), \( E_1 \) is at least equal to \( E_2 \), and possibly greater.
For \( x > -248 \), \( E_2 \) becomes greater.
Thus, the correct statement 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 \( - (12x - 3) + 4(x + 5) - \left(\frac{1}{2}x - 3\right) + 4(x + 5) \).
### Part B
**Explanation & Summary**:
We found each expression’s simplified form and compared them. The relationship changes depending on the value of \( x \). This indicates that one expression does not consistently dominate the other; rather, it varies based on the specified range of \( x \).
### Part C
**New Expression**:
To construct a new expression that is always greater than both:
We can determine a constant value greater than our evaluated constants from \( E_1 \) and \( E_2 \).
Choose a simple constant, e.g.,
\[
E_3 = 100
\]
This expression will always be greater than either of the expressions since both \( 24 < 100 \) and \( 86 < 100 \) are true regardless of \( x \).
Thus, one such valid expression is \( 100 \).