Question

Two friends are planning for a gathering. the food budget is modeled by one half times the absolute value of the quantity x minus 120 end quantity equals 10 comma where x is the amount spent on food. what are the least and greatest amounts that the two friends could spend on food?

1. An absolute value of |x| {modulus of x} is the value of a real number x. The least and greatest amounts that the two friends could spend on food are 100 and 140, respectively.

### What is Absolute Value?

An absolute value of |x| {modulus of x} is the value of a real number x, the value we get is always a non-negative number, for example, |-5| will give 5, and also, |5| will give 5 as well.
Given that the food budget is modeled as (1/2)|x – 120|=10, where x is the amount spent on food.
Since in the given equation for the absolute quantity the critical point is 120. Therefore, we can write,
A.) When the value of x≥120,
(1/2)|x – 120|=10
|x – 120| = 10 × 2
|x – 120| = 20
Now, the value of x is greater than 120, the absolute quantity will be positive,
x – 120 = 20
x = 20 + 120
x = 140
B.) When the value of x<120,
(1/2)|x – 120|=10
|x – 120| = 10 × 2
|x – 120| = 20
Now, the value of x is less than 120, the absolute quantity will be negative,
-(x – 120) = 20
-x + 120 = 20
-x = 20 – 120
-x = -100
x = 100
Hence, the least and greatest amounts that the two friends could spend on food are 100 and 140, respectively.