Compound Inequalities

A compound inequality joins two inequalities with the word “and” or the word “or”. “And” means both must be true at once, like a speed limit with a minimum and a maximum. “Or” means at least one must be true, like a discount for children or for seniors. You solve each piece the way you did in the last lesson, then combine the results: the overlap for “and”, and everything that either piece covers for “or”.

Two inequalities, one joining word

A compound inequality is two inequalities joined by the word and or the word or. A solution is a value of the variable that makes the whole statement true.

• With and, a number must make both inequalities true.
• With or, a number must make at least one of them true. Making both true is allowed. (In math, “or” includes “both”.)

Example with “and”: x > 1 and x < 6. The number 3 is a solution, because 3 > 1 and 3 < 6 are both true. The number 8 is not, because 8 < 6 is false.

Example with “or”: x < 1 or x > 6. The number 0 is a solution, because 0 < 1 is true. The number 8 is a solution, because 8 > 6 is true. The number 3 is not, because 3 < 1 is false and 3 > 6 is false.

Every solution of an “and” statement is also a solution of each piece alone. Every solution of either piece alone is also a solution of the “or” statement. So “and” can only narrow things down (or leave them the same), and “or” can only widen them (or leave them the same).

The three-part (between) form

Many “and” inequalities are written in one line, like −2 < x ≤ 5. This is shorthand for x > −2 and x ≤ 5. You read it as “x is greater than −2 and at most 5”. It describes the numbers between −2 and 5, and the symbols tell you which ends count: −2 does not count (it goes with <), and 5 does count (it goes with ≤).

For a three-part inequality to describe a stretch of numbers, both signs must point the same way, with the smaller number on the left when the signs are < or ≤. In −2 < x ≤ 5 the smaller number is on the left. With the signs pointing the other way, the larger number goes first, as in 5 ≥ x > −2, which says the same thing. A statement like 5 < x < −2 can be written, but it asks for x > 5 and x < −2, and no number does both.

On a number line, draw an open circle at −2, a closed circle at 5, and shade the stretch between them.

In everyday speech, “between” may or may not include the end numbers. In math, the symbols decide, so a problem should say “inclusive” or “including” when the ends count.

Solving an “and” inequality

There are two reliable ways.

Split it. Write two inequalities, solve each one, graph both, and keep the overlap.

Work on all three parts. In a three-part inequality, do the same move to every part. Example: −5 < 2x + 1 ≤ 9. Subtract 1 from all three parts: −6 < 2x ≤ 8. Divide all three parts by 2: −3 < x ≤ 4.

If you multiply or divide all three parts by a negative number, reverse both signs. Example: 4 < 1 − 3x ≤ 13. Subtract 1 from every part: 3 < −3x ≤ 12. Divide every part by −3 and reverse both signs: −1 > x ≥ −4. Now the larger number is on the left, so read the chain from right to left to put the smaller number first: −4 ≤ x < −1. (Writing a chain in the opposite order turns each symbol to face the other way, just as 3 < x becomes x > 3. The meaning does not change.)

Do each move to all three parts, never just the middle.

Solving an “or” inequality

An “or” inequality comes as two separate pieces. Solve each piece on its own, graph both on the same number line, and keep everything that either piece shades.

Example: 2x + 1 < −5 or x − 3 ≥ 2. The first piece gives 2x < −6, so x < −3. The second gives x ≥ 5. The answer is x < −3 or x ≥ 5: two separate rays with a gap between them.

An answer like this cannot be squeezed into one three-part inequality. Writing it as 5 ≤ x < −3 would mean x ≥ 5 and x < −3, which is impossible. Keep the two pieces and the word “or”.

When both rays point the same way, one contains the other. For x > 2 or x > 7, any number that satisfies x > 7 also satisfies x > 2, so the answer is just x > 2. For x > 2 and x > 7, a number needs both, so the answer is x > 7.

Graphing, and interval notation

Graph each piece on its own number line first, using the open and closed circles from the last lesson. Then build the answer line:

• For and, shade only where both pieces are shaded (the overlap).
• For or, shade wherever either piece is shaded.

Interval notation is a short way to write a solution set. A parenthesis means the end is not included, and a square bracket means it is. So −2 < x ≤ 5 is (−2, 5]. The symbol ∪ joins the pieces of an “or”: x < −6 or x ≥ 7 is (−∞, −6) ∪ [7, ∞). Infinity is not a number you can reach, so it always gets a parenthesis. You can solve these problems without interval notation, but you will see it in many books.

No solution, every real number, and one-number answers

Combine the pieces carefully, and expect a few surprises.

• “And” with no overlap: no solution. For x < −1 and x > 2, no number is less than −1 and also greater than 2. Nothing is shaded on the answer line. This is a correct answer, not a mistake.
• “Or” with two rays that point opposite ways and overlap: every real number. For x < 6 or x > −3, a number that is not less than 6 is at least 6, so it is greater than −3. Every number gets in.
• “And” where the pieces meet at one point: exactly one solution. For x ≤ 9 and x ≥ 9, only 9 is both at most 9 and at least 9. The solution is x = 9.
• “Or” that misses one point: every number except that one. For x < 9 or x > 9, the number 9 is neither less than 9 nor greater than 9, so it is the only number left out.

Word problems: “and” or “or”?

Decide between “and” and “or” from the meaning of the story.

• Words like between, from … to …, or at least … but at most … describe a range with two limits that must both hold: “and”.
• Words like either, outside the range, too low or too high, or two separate groups that each get the benefit describe “or”: one condition is enough.

Define the variable in words first. Then write each limit as its own inequality: “at least” is ≥, “at most” is ≤, “more than” is > and “fewer than” is <. Decide whether each end counts. “From 15 to 25 degrees, including both” means 15 ≤ C ≤ 25, but “more than 15 and less than 25” means 15 < C < 25.

Solve with the same moves, and say what the answer means in the story, with units. If only whole numbers make sense, list or count the whole numbers that fit.