Question

Given `-2x-5 < -2` I got `-2x < 3`. Is this correct? How do I solve from here?

Answer 1

Your answer is not wrong; but it is incomplete.
`color(white)("XXX")-2x-5 < -2color(white)("XX")rarrcolor(white)("XX")color(green)(x > -3/2)`

Explanation:

You managed to get from
`color(white)("XXX")-2x-5 < -2`
to
`color(white)("XXX")-2x < 3`
So let's start from that point.

In "solving" one of these problems the goal is to isolate a single `x` on one side of the inequality.

I see 2 ways of doing this:

Method 1
Remember the rule that tells you that you can multiply or divide both sides of an inequality by any negative number if you reverse the inequality sign.

`color(white)("XXX")-2x color(blue)< 3`
`color(white)("XXXXX")`after dividing both sides by `(-2)`, becomes
`color(white)("XXX")x color(red)( > ) -3/2`

Method 2
Remember that you can add or subtract the same amount to both sides of an inequality
or multiply or divide both sides by any amount greater than zero
without effecting the orientation of the inequality.

`color(white)("XXX")-2x < 3`
`color(white)("XXXXXX")`after adding `2x` to both sides, becomes
`color(white)("XXX")0 < 3 +2x`
`color(white)("XXXXXX")`then subtracting `3` from both sides, becomes
`color(white)("XXX")-3 < 2x`
`color(white)("XXXXXX")`and, finally, dividing both sides by `2`
`color(white)("XXX")-3/2 < x`
`color(white)("XXXXXXXXX")` which is just another way of writing `x > -3/2`

Answer 2

`x > -3/2`

Explanation:

`-2x-5<-2`

Rule of thumb worth remembering, if you multiply each side of an inequality by a negative number you must switch the direction of the inequality sign (because the sign of each side has changed so you are, in crude terms, looking at a reflection about the origin of the original relationship on the number line).

So, multiplying by `-1`, it becomes:

`2x+5 color(red)(>) 2`

Then some algebra:

`2x+5 - 5 > 2 - 5`

`2x > -3`

And divide each side by 2:

`x > -3/2`

Doing it just by adding/ subtracting the right things on each side of the inequality is a bit slower but here it is.

`-2x color(red)(+ 2x) -5 color(blue)(+ 5)< -2 color(red)(+ 2x) color(blue)(+ 5)`

`implies 0< 3 + 2x`

`0 - 3< 3 - 3 + 2x`

`- 3< 2x`

Switch the order:

`2x > - 3`

And divide each side by 2:

`x > -3/2`

Perhaps the most important bit: You can always check your answer too, eg solve the equality first, or sub in values either side of where you think the inequality comes out. Or just sketch/ plot it.