Step by step
The question
Question: solve x + 3 ≤ 0 or x + 3 ≥ 7.
Two inequalities, joined by "or".
"Or" means: x can make one of them true, or the other.
We solve both, side by side.
1 Remove the + 3
We want x by itself.
The + 3 is in the way.
In each one, we take away 3 on both sides.
+ 3 and − 3 make zero, in both.
0 minus 3 is −3.
7 minus 3 is 4.
So x ≤ −3 or x ≥ 4.
2 Show it on a number line
Books also write the answer like this.
It is called interval notation.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
3 Check
Let's test one number from each part.
First −4, from the left part.
−1 is less than or equal to 0.
True!
Now 5, from the right part.
8 is greater than or equal to 7.
True!
Both parts work!
Let's look back.
First: remove the + 3.
Then: show it on a number line.
Last: check.