Question

How do you solve `10x-6<=4x+42` and graph the solution on a number line?

Answer 1

`x le 8`

Explanation:

First, subtract `4x` from both sides.

`10x - 6 - color(blue)(4x) le 4x + 42 - color(blue)(4x)`

`6x - 6 le cancel(4x) + 42 - cancel(color(blue)(4x)`

`6x - 6 le 42`

Next, add 6 to both sides.

`6x - 6 + color(red)6 le 42 + color(red)6`

`6x - cancel6 + cancelcolor(red)6 le 48`

`6x le 48`

Finally, divide both sides by 6.

`(6x)/color(limegreen)6 le 48/color(limegreen)6`

`(cancel6x)/cancelcolor(limegreen)6 le 8`

`x le 8`

This is the solution! To graph it on a number line, first consider what the solution is telling you: "x is less than or equal to 8"

We need to include 8 in our solution, and everything less than 8. Therefore, we will draw a solid dot at `x=8` on our number line, and then shade in everything to the left of it. The number line will look like this:

Final Answer

Answer 2

See a solution process below:

Explanation:

First, add `color(red)(6)` and subtract `color(blue)(4x)` from each side of the inequality to isolate the `x` term while keeping the inequality balanced:

`-color(blue)(4x) + 10x - 6 + color(red)(6) <= -color(blue)(4x) + 4x + 42 + color(red)(6)`

`(-color(blue)(4) + 10)x - 0 <= 0 + 48`

`6x <= 48`

Now, divide each side of the inequality by `color(red)(6)` to solve for `x` while keeping the inequality balanced:

`(6x)/color(red)(6) <= 48/color(red)(6)`

`(color(red)(cancel(color(black)(6)))x)/cancel(color(red)(6)) <= 8`

`x <= 8`

To graph this we draw a solid circle at `8` on the number line. The circle is solid because the inequality operator contains an "or equal to" clause.

We will draw an arrow to the left of `8` because the inequality operator also contains a "less than" clause: