The strategy for an "or" inequality is to divide and conquer. We'll split the inequalities up, shake 'em down, solve them, and then stitch the resulting mess back up.
5x > 2x + 18
3x > 18
x > 6
We have nothing special for the first one. Onward to the second.
-4x – 1 < 19
-4x < 20
Beep boop, negative division imminent. Deploy the reverse sign.
x > -5
We're not done yet; we have to combine our results. The solutions to the inequality are:
x > -5 or x > 6
Hold up, we can actually simplify this a little more. Both inequality signs point in the same direction. So we can drop x > 6 altogether.
See, x > -5 does everything x > 6 does, and then some. So, x > -5 is our solution. What a hard worker. Why can't you be like that, x > 6?
Example 2
Solve x > 3x + 8 or 7x ≤ 8x – 6.
Like King Solomon, we're going to pretend to cut this baby in half, but we'll make it all better in the end.
x > 3x + 8
-8 > 2x
-4 > x
x < -4
Just…ignore the fact we've actually split the inequality in half. We'll fix it soon, promise.
7x ≤ 8x – 6
6 ≤ x
x ≥ 6
Now we have x < -4 or x ≥ 6 as our solution. It's totally still in one piece. We'd like to see Solomon try to do that. With an inequality, of course, not a baby.
Example 3
Solve 17 ≤ 5x – 3 < 22.
This inequality is all set up for us to solve. We just need to remember to apply all of our operations to all three parts, like adding 3:
20 ≤ 5x < 25
The coefficient is positive, so we're in the clear.
4 ≤ x < 5
Our solution can include 4, but doesn't include 5. Sorry, 5, but you're not invited. (We're not actually sorry at all.)
