Question

martin is considering the expressions 1/2(7x+48) and -(1/2x-3). he wants to know if one expression is greater than the other for all values of x. which statement about the relationship between the expressions is true?

Answer 1

The correct answer is that the expressions are equal when x=-5.25; when x>-5.25, the first expression is greater; and when x<-5.25, the first expression is smaller.

Explanation:
We set the expressions equal to one another first:
1/2(7x+48)=-(1/2x-3).

Using the distributive property on both sides, we have
(1/2)*7x+(1/2)*48=-1/2x- -3
7/2x+24=-1/2x+3.

To solve this, we can add 1/2x to both sides:
7/2x+24+1/2x=-1/2x+3+1/2x
8/2x+24=3
4x+24=3.

Subtract 24 from both sides:
4x+24-24=3-24
4x=-21.

Divide both sides by 4:
4x/4=-21/4
x=-21/4=-5.25.

This is the point where the two expressions are equal. If we were to repeat this process, except use a > symbol rather than =, we would have the solution x>-5.25; this means when x>-5.25, the first expression is larger.

Similarly, we can replace the equals with a less than, and end up with x<-5.25, which means when x is below this number, the first expression is smaller than the second.