Hello, Rachel!
When solving compound inequalities, you solve them normally first. For the first equations, −9x+5≤17 or 13x+25≤−1, we will solve each one individually.
−9x+5≤17 or 13x+25≤−1
-5 -5 -25 -25
--------------------------------------
-9x≤12 13x ≤-26
------------------------------
-9 13
x≥-4/3 or x≤-2
When you think about this on a number line, the solution makes sense because how could x be BOTH greater than -4/3 AND less than -2? It's not possible!
So now, how do we write the solutions in interval notation? Again, imagining a number line, x is all values less than -2 or all values greater than -4/3. When there is an "or", you use a ∪ (union) symbol. We should work from least to greatest. The values less than -2 would be negative infinity and then continue up to and equal -2. To show that x can equal that value/solution, we use a bracket. You don't use a bracket for infinity because how can you touch infinity?
(-∞, -2]
Then, we put a ∪ (union) to show we're not done! There are more possible solutions. The solutions pick back up again at -4/3, and x can equal -4/3, so we use a bracket. Then, the solutions continue to positive infinity.
(-∞,-2]∪[-4/3,∞)
−15x+60≤105 and 14x+11≤−31
The only difference in these two inequalities is that the solutions must fit BOTH requirements.
-15x+60≤105 14x + 11 ≤-31
-60 -60 -11 -11
-------------------- --------------------
-15x ≤ 45 14x ≤ -42
-------------- ------------
-15 14
x≥-3 x≥-3
Strangely, both solutions are greater than -3, which would be written as [-3,∞).
However, just to show you what would happen if we had gotten a different answer, let's assume that for one of the solutions, you got x≤14 (and the other solution remained at x≥-3).
To write the solution in interval notation, you would start from lowest possible solution to highest possible solution. The lowest possible solution is -3 and the highest possible solution is 14. If you think about it, we want the solutions to be BOTH greater than and equal to -3 AND less than 14. Is this possible? Yes! A number like 5 is greater than -3 but less than 14. If it was NOT possible, then your answer would be no solution!
For this made-up example, [-3,14] would be the answer in interval notation.
