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Algebra I

Compound Inequalities

Lesson

A compound inequalityis two inequalities joined together. The most common kind is the “between” or AND case, which says that xx is between two values:

−3<x<5-3 < x < 5

This reads “xx is greater than −3-3 AND less than 55.”

To solve, do the same operation to all three parts at once. Treat the middle as the target you want to isolate.

The same rules from one-step inequalities apply: when you multiply or divide all parts by a negative, you must flip both inequality symbols.

Worked example 1

−2<x+3<7-2 < x + 3 < 7

Subtract 3 from all three parts:

−5<x<4-5 < x < 4

Worked example 2

4≤2x≤104 \le 2x \le 10

Divide all three parts by 2 (positive — no flip):

2≤x≤52 \le x \le 5

How to type your answer

Write your final answer in the form a<x<b (or with <=). No spaces. Examples: -5<x<4, 2<=x<=5, -1<x<=3.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

1<x+2<51 < x + 2 < 5

Problem 2

0<x−1<40 < x - 1 < 4

Problem 3

2≤2x≤82 \le 2x \le 8

Problem 4

−3<x+1<3-3 < x + 1 < 3

Practice

Standard problems matching the lesson.

Problem 5

−2<x+3<7-2 < x + 3 < 7

Problem 6

1≤x−4≤61 \le x - 4 \le 6

Problem 7

−6<2x<8-6 < 2x < 8

Problem 8

4≤2x≤104 \le 2x \le 10

Problem 9

−3≤x+2≤5-3 \le x + 2 \le 5

Problem 10

0<x−7<90 < x - 7 < 9

Problem 11

−6≤3x≤12-6 \le 3x \le 12

Problem 12

5<x+1≤105 < x + 1 \le 10

Problem 13

−8≤2x−4≤6-8 \le 2x - 4 \le 6

Problem 14

1<3x+4<131 < 3x + 4 < 13

Problem 15

0≤4x+8≤160 \le 4x + 8 \le 16

Problem 16

−7<5x−2<13-7 < 5x - 2 < 13

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

−6<−2x<4-6 < -2x < 4

Problem 18

−9≤−3x≤6-9 \le -3x \le 6

Problem 19

−15≤3x−6≤9-15 \le 3x - 6 \le 9

Problem 20

1<2x+3≤111 < 2x + 3 \le 11

Problem 21

−4≤−(x−2)≤5-4 \le -(x - 2) \le 5

Problem 22

8<4x≤208 < 4x \le 20

Practice

Standard problems matching the lesson.

Problem 23

A movie theater's kids' discount is for ages over 5 and under 13. Let x be the age. Write the inequality.

Problem 24

A safe driving speed is at least 25 mph and no more than 55 mph. Let x be the speed (mph). Write the inequality.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

A roller coaster requires riders taller than 48 in and no taller than 78 in. Let x be the height (in). Write the inequality.

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