Compound Inequality

Q: Show the answer

First: $2x < -4$, so $x < -2$. Second: $3x \ge 12$, so $x \ge 4$. Answer: $x < -2$ or $x \ge 4$.

Q: Show the answer

$12 \le c \le 20$, where c is the cost in dollars.

Compound Inequality: meaning, method and solved example (Algebra 1) – Maths PDF

Sommaire

In short: A compound inequality ties two inequality statements together with the connector and or or, so a number belongs to the solution when it meets both conditions or at least one.

Algebra 1AlgebraA-REI.B.3

What is Compound Inequality?

A compound inequality packs two conditions into a single statement, in the way a rule such as “at least 65 but under 80” does. With the connector and, a number must pass both tests, which gives a segment of the number line, as in $-2 \le x < 5$. With or, a number may pass either test, which gives two separate rays, as in $x < -3$ or $x \ge 4$.

Solve each part with the usual rules, and remember to flip the inequality sign whenever you multiply or divide both sides by a negative number. Then combine the two answers according to the connector: overlap for and, union for or.

How to spot it

A quantity must stay between two limits, such as a greenhouse kept at least 65 degrees but under 80 degrees.
The problem contains a connecting word like and, or, between or either, linking two conditions on one variable.
A variable is sandwiched between two inequality signs, like $-7 < 3x - 1 \le 14$, and must be isolated in the middle.
A graph shows one segment with two endpoints, or two rays pointing away from each other along the number line.

The method

Decide whether the connector is and (both statements must be true) or or (at least one must be true).
Solve each inequality separately, or do the same operation on all three parts of a sandwiched one.
Graph each solution, using a filled circle for $\le$ or $\ge$ and an open circle for $<$ or $>$.
For and keep only the overlap of the graphs; for or keep everything covered by either graph.

The graph of -2 ≤ x < 5: every number from -2 up to, but not including, 5.

Compound Inequality: solved example

Problem

Solve the compound inequality $-7 < 3x - 1 \le 14$ and describe the solution.

Solution

Add 1 to all three parts: $-6 < 3x \le 15$. Divide all three parts by 3 (a positive number, so the signs stay): $-2 < x \le 5$.

The sandwich form works because the same operation hits the left, middle and right parts, so the order of the three stays correct. The solution is every real number greater than $-2$ and at most 5: $-2 < x \le 5$, an open circle at $-2$ and a closed circle at 5.

Try it

Question 1

Solve: $2x + 5 < 1$ or $3x - 4 \ge 8$.

Show the answer

First: $2x < -4$, so $x < -2$. Second: $3x \ge 12$, so $x \ge 4$. Answer: $x < -2$ or $x \ge 4$.

Question 2

Write an inequality for a lunch order that costs at least $12 but no more than $20.

Show the answer

$12 \le c \le 20$, where c is the cost in dollars.

Compound Inequality vs. System of Inequalities

A compound inequality has one variable and lives on a number line. A system of inequalities usually has two variables and its solution is a shaded region of the coordinate plane. Both ask for values that satisfy several conditions at once, but the picture, the number of variables and the way you graph the answer are different.

System of Inequalities

Keep going

Lessons

Linear Inequalities in One Variable

Worksheets and quizzes

Related words

Whole dictionary: A–Z · Algebra 1 · Algebra

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