Question

How do you solve `-8(4x+6)< -24` and graph the solution on a number line?

Answer 1

`x>(-3/4)`

Explanation:

Distribute the `-8` into `(4x+6)`.
`-32x-48<-24`

Add `-48` on both sides.
`-32x<24`

IMPORTANT: Since you're dividing both sides by a negative number (`-32`), the`<`will become a`>` ---> `x>(-24/32)`

After simplifying the fraction, you will end up with `x>(-3/4)`.

How To Graph the Inequality
- Draw a number line such that it includes `-3/4`. You can count by one-fourths.
- Draw a point on `-3/4`. It should be an open (not filled) point because the inequality does not include `-3/4` as an answer; in other words, any value of x HAS to be greater than `-3/4`.
It would only be a closed point if the inequality is `x>=(-3/4)`.
- Draw a straight arrow pointing to the right that connects to the point at `-3/4`.

I hope this helps a lot! :)

Answer 2

See explanation

Explanation:

Given: `-8(4x+6) < -24`

Two approaches the first step:

`color(brown)("Approach 1")`

Multiply both sides by (-1) to make everything positive and turn the inequality sign the other way round.

`+8(4x+6) > +24larr` the wide part of > points to `8(4x+6)`

`color(brown)("Approach 2")`

Given: `-8(4x+6) < -24`
As in the algebra shortcut method of changing sides of the = sign.

Move what is on the left of the `<` to its right and move what is on the right of < to its left. In doing so change their signs

`+24<+8(4x+6)larr` the wide part of > points to `8(4x+6)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I choose the form: `+8(4x+6) > +24`

Divide both sides by 8

`4x+6>3`

Subtract 6 from both sides

`4x > -3`

Divide both sides by 4

`x> -3/4`

Number line 'graph'. The circle is hollow indicating 'greater than'.

Suppose it had been 'greater then or equal too'. In this case the circle would be filled in.
Tony B

What is actually happening: The colored in area below `y=-24` is the feasible solution area for `(x,y)`

Tony B `x> -3/4`