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

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-9x≤12 13x ≤-26

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-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,∞)__

−15*x*+60≤105 and 14*x*+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

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-15x ≤ 45 14x ≤ -42

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-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.