Question

How do you solve and write the following in interval notation: `8x -3x + 2< 2(x + 7)`?

Answer 1

`(-oo,4)`

Explanation:

We have the inequality

`8x-3x+2 < 2(x+7)`

On the left hand side, the `8x` and `-3x` can be combined:

`overbrace(8x-3x)^(8x-3x=5x)+2 < 2(x+7)`

`5x+2 < 2(x+7)`

Next, on the right hand side, distribute the `2`:

`5x+2 < overbrace(2(x+7))^(2(x)+2(7))`

`5x+2 < 2x+14`

Subtract `2x` from both sides.

`overbrace(5x-2x)^(3x)+2 < overbrace(2x-2x)^0+14`

`3x+2 < 14`

Subtract `2` from both sides.

`3x+overbrace(2-2)^0 < overbrace(14-2)^12`

`3x < 12`

Divide both sides by `3`.

`(3x)/3 < 12/3`

`x < 4`

Now, we need to express `x<4` in interval notation. Since `x` has to be less than `4`, we know that `4` is the upper bound of the interval.

Since there is no lower bound, `x` can extend all the way down, from `4`, to `0`, into the negative numbers, working its way left on the number line all the way to negative infinity.

So, the interval is `(-oo,4)`. Note that `(` and `)` are used instead of `[` and `]` because `x<4`, not `x<=4`. Additionally, `(` and `)` are always used with `oo` because `x` can never equal `oo`.