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Compound Inequality

Compound Inequality

Two or more inequalities taken together. Often this refers to a connected chain of inequalities, such as 3 < x < 5.

Formally, a compound inequality is a conjunction of two or more inequalities.

 

See also

Compound fraction

Worked Example

Problem: Solve the compound inequality −2≤2x+4<10-2 \le 2x + 4 < 10 and graph the solution.
Step 1: Recognize that this is an 'and' compound inequality. It means −2≤2x+4-2 \le 2x + 4 AND 2x+4<102x + 4 < 10. You can solve both parts at once by performing the same operations on all three sides.
−2≤2x+4<10-2 \le 2x + 4 < 10
Step 2: Subtract 4 from all three parts to begin isolating xx.
−2−4≤2x+4−4<10−4-2 - 4 \le 2x + 4 - 4 < 10 - 4
Step 3: Simplify each part.
−6≤2x<6-6 \le 2x < 6
Step 4: Divide all three parts by 2. Since 2 is positive, the inequality signs stay the same.
−3≤x<3-3 \le x < 3
Step 5: Write the solution in interval notation. The bracket at −3-3 means it is included; the parenthesis at 33 means it is excluded.
[−3,3)[-3, 3)
Answer: The solution is −3≤x<3-3 \le x < 3, or in interval notation [−3,3)[-3, 3). On a number line, shade from −3-3 (closed dot) to 33 (open dot).

Another Example

Problem: Solve the compound inequality 3x−1<−73x - 1 < -7 or 2x+5≥112x + 5 \ge 11.
Step 1: This is an 'or' compound inequality, so solve each part separately. Start with the first inequality.
3x−1<−73x - 1 < -7
Step 2: Add 1 to both sides, then divide by 3.
3x<−6  ⟹  x<−23x < -6 \implies x < -2
Step 3: Now solve the second inequality. Subtract 5 from both sides, then divide by 2.
2x+5≥11  ⟹  2x≥6  ⟹  x≥32x + 5 \ge 11 \implies 2x \ge 6 \implies x \ge 3
Step 4: Combine the results with 'or'. The solution is the union of both sets.
x<−2orx≥3x < -2 \quad \text{or} \quad x \ge 3
Answer: The solution is x<−2x < -2 or x≥3x \ge 3, which in interval notation is (−∞,−2)∪[3,∞)(-\infty, -2) \cup [3, \infty). On a number line, shade to the left of −2-2 (open dot) and to the right of 33 (closed dot).

Frequently Asked Questions

What is the difference between 'and' and 'or' compound inequalities?
An 'and' compound inequality requires both conditions to be true at the same time, so the solution is the overlap (intersection) of the two individual solutions. An 'or' compound inequality requires at least one condition to be true, so the solution is the combination (union) of both individual solutions. 'And' typically produces a bounded interval like −3≤x<3-3 \le x < 3, while 'or' often produces two separate rays extending in opposite directions.
Can a compound inequality have no solution?
Yes. An 'and' compound inequality has no solution when the two conditions don't overlap. For example, x>5x > 5 and x<2x < 2 has no solution because no number is simultaneously greater than 5 and less than 2. An 'or' compound inequality, however, almost always has a solution because you only need one condition to hold.

'And' compound inequality vs. 'Or' compound inequality

An 'and' compound inequality finds the intersection of two solution sets — values that satisfy both inequalities. An 'or' compound inequality finds the union — values that satisfy at least one inequality. On a number line, 'and' solutions appear as a single connected segment (or nothing), while 'or' solutions often appear as two separate regions.

Why It Matters

Compound inequalities model real-world situations with upper and lower bounds. For instance, a safe temperature range for storing medicine might be 2≤T≤82 \le T \le 8 degrees Celsius — a compound inequality in disguise. They also appear frequently in absolute value problems, where ∣x∣<5|x| < 5 translates to the compound inequality −5<x<5-5 < x < 5.

Common Mistakes

Mistake: Flipping the inequality sign when multiplying or dividing all parts by a negative number in only some of the parts.
Correction: When you multiply or divide a compound inequality by a negative number, you must reverse every inequality sign and reverse the order of the entire chain. For example, multiplying −6≤−2x<4-6 \le -2x < 4 by −12-\frac{1}{2} gives −2<x≤3-2 < x \le 3 (all signs flip).
Mistake: Writing an 'or' inequality as a single chain, such as 5<x<−25 < x < -2.
Correction: A single chain like a<x<ba < x < b always implies 'and'. If the solution is x>5x > 5 or x<−2x < -2, you cannot collapse it into one chain. Write it as two separate inequalities joined by the word 'or'.

Related Terms

  • Inequality — A single inequality; building block of compounds
  • Conjunction — Logical 'and' connecting two statements
  • Disjunction — Logical 'or' connecting two statements
  • Absolute Value — Absolute value inequalities become compound inequalities
  • Interval Notation — Compact way to express compound inequality solutions
  • Intersection — Solution set of an 'and' compound inequality
  • Union — Solution set of an 'or' compound inequality