Question

The largest set of x values satisfying

2018x−p<2020x+p and 7x+3p<10x−2
can be written in simplest form as x>mn for positive m and n. What is the value of m+n?

Answer 1

If
`p\ge -(1)/(3),` then
`x>(2)/(3)+p` and
`mn=(2)/(3)+p=\left((2)/(3)+p\right)\cdot 1,\ \ m+n=(5)/(3)+p`

Explanation:

Solve two inequalities for x.

1.
`2,018x-p<2,020x+p`

Separate terms with x and without x into two sides:

`2,018x-2,020x<p+p\\ \\-2x<2p`

Multiply by -1:

`2x>-2p\\ \\x>-p`

2.
`7x+3p<10x-2`

Separate terms with x and without x into two sides:

`7x-10x<-2-3p\\ \\-3x<-2-3p`

Multiply by -1:

`3x>2+3p\\ \\x>(2)/(3)+p`

Find the largest set of x values satisfying both inequalities:

`x>-p\\ \\x>(2)/(3)+p`

If

`-p>(2)/(3)+p\\ \\-2p>(2)/(3)\\ \\2p<-(2)/(3)\\ \\p<-(1)/(3),`

then
`x>-p` and
`mn=-p=p\cdot (-1),\ m+n=p-1.` In this case both m and n are negative.

If
`p>-(1)/(3),` then
`x>(2)/(3)+p` and
`mn=(2)/(3)+p=\left((2)/(3)+p\right)\cdot 1,\ \ m+n=(5)/(3)+p`

If
`p=-(1)/(3),` then
`x>(1)/(3)` and
`mn=(1)/(3)=(1)/(3)\cdot 1,\ \ m+n=(4)/(3)`