Question

How do you solve `\frac { 7} { 5} x - 5\geq \frac { 1} { 6} x + 1`?

Answer 1

We need to isolate the variable, `x`, but first, let's get the constants on one side

`7/5x - 5 >= 1/6x + 1`

add `5` to both sides

`7/5x >= 1/6x + 6`

subtract `1/6x` on both sides

`7/5x - 1/6x >= 6`

`6/6 xx 7/5x - 1/6x xx 5/5 >= 6`

`42/30x - 5/30x >= 6`

`37/30x >= 6`

divide by `37/30` on both sides

`x >= 6 -: 37/30`

multiply by the reciprocal

`x >= 6/1 xx 30/37`

`x >= 180/37 ~~ 4.86`

`* * * * * * * * * * * * * * * * *`

Let's check our work by solving the original equation, using `180/37` instead of `x`. If we were right, both sides of the equation should look the same.

`7/5x - 5 >= 1/6x + 1`

`7/5 (180/37) - 5 >= 1/6 (180/37) + 1`

`252/37 - 5 >= 30/37 +1`

`67/37 >= 67/37`

We were right!

`x` is `180/37`

Answer 2

`x >=4 37/180`

Explanation:

Treat an inequality in the same way as an equation, unless you multiply or divide by a negative number, (then the sign changes around.)

`7/5x-5 >= 1/6x+1" "` multiply by 30 to cancel the denominators

`(color(blue)(cancel30^6xx)7)/cancel5x-color(blue)(30xx)5 >=(color(blue)(cancel30^5xx) 1)/cancel6x+color(blue)(30xx)1`

`42x-150 >= 5x+30" "larr` re-arrange the terms

`42x -5x >= 30 +150`

`37x >= 180`

`x >= 180/37`

`x >= 4 37/180`