Question

The surface area S of a cube with edge length x is given by
S(x) = 6x2
for x > 0. Suppose the cubes your company manufactures are supposed to have a surface area of exactly 42 square centimeters, but the machines you own are old and cannot always make a cube with the precise surface area desired. Write an inequality using absolute value that says the surface area of a given cube is no more than 3 square centimeters away (high or low) from the target of 42 square centimeters.

Answer 1

the surface area of a given cube is no more than 3 square centimeters away (high or low) from the target of 42 square centimeters.'' is equivalent to saying that the surface area of a cube is within 3cm² of 42cm² ... i.e. the difference between the actual surface area of a cube and the target of 42cm² is less than or equal to 3cm² ... i.e. │S(x) − 42│≤ 3

Therefore,
│S(x) − 42│≤ 3   ⇒   │6x² − 42│ ≤  3       
                                                                      
                                 │6(x² − 7)│ ≤  3
                                    6│x² − 7│ ≤  3
                                     │x² − 7│ ≤  ½