(a)

Question

Martin is considering the expressions 1/2 ( 7x +48) - ( 1/2 x - 3) + 4 (x + 5)
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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Answer 1

Part A

To evaluate the relationship between the given expressions, let's first simplify both expressions step-by-step.

1. First expression: \[ \frac{1}{2}(7x + 48) - \left(\frac{1}{2}x - 3\right) + 4(x + 5) \] Distributing each term: \[ = \frac{7}{2}x + 24 - \frac{1}{2}x + 3 + 4x + 20 \]

   Now, simplify: \[ = \left(\frac{7}{2} - \frac{1}{2} + 4\right)x + (24 + 3 + 20) \] \[ = \left(\frac{6}{2} + 4\right)x + 47 \] \[ = (3 + 4)x + 47 = 7x + 47 \]

2. Second expression: \[ -\left(\frac{1}{2}x - 3\right) + 4(x + 5) \] Distributing: \[ = -\frac{1}{2}x + 3 + 4x + 20 \]

   Now, simplify: \[ = \left(-\frac{1}{2} + 4\right)x + (3 + 20) \] \[ = \left(-\frac{1}{2} + \frac{8}{2}\right)x + 23 \] \[ = \frac{7}{2}x + 23 \]

At this point, we can compare:

• First expression: \( 7x + 47 \)
• Second expression: \( \frac{7}{2}x + 23 \)

Determine Relationship:

To see if \(7x + 47\) is greater than, less than, or equal to \(\frac{7}{2}x + 23\):

Set up the inequality:

\[ 7x + 47 > \frac{7}{2}x + 23 \]

Subtract \(\frac{7}{2}x\) from both sides:

\[ 7x - \frac{7}{2}x + 47 > 23 \]

Convert \(7x\) to fractions:

\[ \frac{14}{2}x - \frac{7}{2}x + 47 > 23 \]

Combine like terms:

\[ \frac{7}{2}x + 47 > 23 \]

Now subtract \(47\) from both sides:

\[ \frac{7}{2}x > 23 - 47 \] \[ \frac{7}{2}x > -24 \]

This means:

\[ x > -\frac{24 \times 2}{7} \] \[ x > -\frac{48}{7} \]

Since this is a condition for \(x\) and does not apply for all \(x\), it indicates that there are scenarios where the first expression can be greater than or less than the second expression.

Therefore, the correct answer to Part A 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 \(-(\frac{1}{2}x - 3) + 4(x + 5)\).

Part B

To explain this:

• I simplified both expressions step-by-step.
• I compared the resulting expressions to see if one is consistently greater or occasionally greater by solving the inequality.
• I found that the relationship depends on the value of \(x\) which leads to the conclusion in Part A.

Part C

To create a new expression that is always greater than both previously evaluated expressions, we could add a constant greater than 0 to one of the expressions:

For example: \[ 8x + 50 \] This new expression \(8x + 50\) is always greater than \(7x + 47\) and \(\frac{7}{2}x + 23\) for all values of \(x\).

New expression: \(8x + 50\)