Question

How do you solve the inequality: `5x + 10 >=10` and `7x - 7 <=14`?

Answer 1

`" "`
Required Solution: `color(blue)(0<=x<=3`

Interval Notation: `color(blue)([ 0,3 ]`

Explanation:

`" "`
The Compound Inequality Expression given:

`color(red)(5x+10>=10 and 7x-7<=14`

`color(green)("Step 1"`

Consider `5x+10>=10` first and simplify.

Subtract `color(red)(10` to both sides of the inequality to obtain

`5x+10- color(red)(10) >=10 - color(red)(10)`

`5x+cancel 10- color(red)(cancel 10) >=cancel 10 - color(red)(cancel 10)`

`5x>=0`

Divide both sides of the inequality by `color(red)(5)`

`(5x)/color(red)(5)>=0/color(red)(5)`

`(cancel 5x)/color(red)(cancel 5)>=0/color(red)(5)`

`color(blue)(x>=0` Intermediate Solution 1

`color(green)("Step 2"`

Consider `7x-7<=14` next.

Add `color(red)(7)` to both sides of the inequality to get

`7x-7+color(red)(7)<=14+color(red)(7)`

`7x-cancel 7+color(red)(cancel 7)<=14+color(red)(7)`

`7x<=21`

Divide both sides of the inequality by `color(red)(7`

`(7x)/color(red)(7)<=21/color(red)(7`

`(cancel 7x)/color(red)(cancel 7)<=cancel 21^color(red)3/color(red)(cancel 7`

`color(blue)(x<=3` Intermediate Solution 2

`color(green)("Step 3"`

Combine both the Intermediate Solution to obtain:

`color(blue)(x>=0 and color(blue)(x<=3`

`color(red)(0<=x<=3` The required solution

Using the interval notation: `color(red)([0,3]`

`color(green)("Step 4"`

You can verify the results using a graph

The first graph below is created using the two inequality expressions given:

`color(red)(5x+10>=10 and 7x-7<=14`

The solution is the shaded common region

You can also graph just the solution to obtain a solution graph:

`color(red)(0<=x<=3`

Hope you find this solution useful.