Inequalities using addition and subtraction

Adding or subtracting the same amount on both sides of an inequality never flips it.
If one amount is bigger than another, adding the same thing to both keeps it bigger.
So if $a > b$, then $a + 5 > b + 5$ and $a - 5 > b - 5$.

You use this to isolate the variable.
Move constant terms away from the variable by adding or subtracting them on both sides.
In $-5x + 2 > 2x + 3$, subtracting $2x$ and then $2$ from both sides gives $-7x > 1$.

For a three-part inequality, do the same thing to all three parts at once.
From $-2 \le -x < 3$, adding $1$ to each part gives $-1 \le 1 - x < 4$.
Every part changes the same way, and the signs stay the same.

Notice that addition and subtraction only slide the numbers; they never reverse the inequality.
That is different from multiplying or dividing by a negative, which does flip it.
So for now, adding and subtracting are completely safe.

Solve $x - 7 \ge 2$.
Add $7$ to both sides: $x - 7 + 7 \ge 2 + 7$.
So $x \ge 9$, and the sign does not change.

Solve $3x - 7 < 2x + 4$.
Subtract $2x$ from both sides, then add $7$: $3x - 2x < 4 + 7$.
So $x < 11$.

Solve the three-part inequality $1 \le x + 3 \le 8$.
Subtract $3$ from every part: $1 - 3 \le x + 3 - 3 \le 8 - 3$.
So $-2 \le x \le 5$.