5-3 Solving Compound Inequalities

Property

A compound inequality consists of two inequalities joined by the logical word "and" or "or".

• 'And' Inequalities: Represent an intersection (overlap). A solution must satisfy BOTH inequalities simultaneously.
• 'Or' Inequalities: Represent a union (combination). A solution must satisfy AT LEAST ONE of the inequalities.

Examples

• "And" Example: Solve $x > 2$ and $x < 7$. The solution is the overlap of these two conditions, which can be written compactly as $2 < x < 7$.
• "Or" Example: Solve $y < -1$ or $y \geq 5$. The solution consists of two completely separate sets of numbers. There is no overlap.
• Error Analysis: For the problem $x \geq -1$ and $x \leq 4$, a student incorrectly shades everything outside the numbers -1 and 4. This is an error because "and" requires an overlap. The correct answer is only the space between the numbers: $-1 \leq x \leq 4$.

Explanation

A compound inequality is a two-part rule. Think of an "and" statement like needing a concert ticket AND a valid ID to enter a venue; you must pass both tests at the same time. Think of an "or" statement like getting a discount if you are a student OR a senior citizen; passing just one of the tests is enough. When solving, always double-check which connector word is used, as it completely changes the final answer!

Property

To solve an "and" compound inequality, solve each inequality separately and find the intersection (overlapping region) of their solution sets.

A chain inequality (e.g., $a < x < b$) is a compact "and" statement. To solve a chain inequality, perform the same inverse operations to all three parts (left, middle, right) simultaneously to isolate the variable in the center.

Examples

• Solving Separate "And" Statements: Solve $4x - 1 < 7$ and $x + 5 \geq 2$.

First, solve each part: $4x < 8 \rightarrow x < 2$, and $x \geq -3$.
The solution is the overlapping region where numbers are both greater than or equal to -3 and less than 2: $-3 \leq x < 2$.

• Solving a Chain Inequality: Solve $-1 < 2x + 3 < 9$.

Subtract 3 from all three parts: $-4 < 2x < 6$.
Divide all three parts by 2: $-2 < x < 3$.

• Chain Inequality with a Negative: Solve $-6 < -2x + 4 < 2$.

Subtract 4 from all parts: $-10 < -2x < -2$.
Divide by -2 and flip all inequality signs: $5 > x > 1$, which is logically rewritten from least to greatest as $1 < x < 5$.

Property

To solve an "or" compound inequality, you must solve the two inequalities completely separately. The final solution is the union of the two solution sets, meaning any number that makes either the first inequality true, the second true, or both true is included in the final answer.

Examples

• Standard "Or" Solution: Solve $3x - 2 > 10$ or $x + 1 < 0$.

Solve each part independently: $3x > 12 \rightarrow x > 4$, and $x < -1$.
The solution is $x < -1$ or $x > 4$.

• Overlapping "Or" Solution: Solve $x - 5 > 2$ or $x - 5 > 0$.

This simplifies to $x > 7$ or $x > 5$. Since any number greater than 7 is already greater than 5, the two rules merge, and the final combined solution is simply $x > 5$.

• All Real Numbers: Solve $x + 3 \leq 5$ or $x - 4 \geq -2$.

This simplifies to $x \leq 2$ or $x \geq 2$. Because this covers all numbers less than 2, equal to 2, and greater than 2, the solution is All Real Numbers.

Explanation

Solving an "or" inequality is about gathering all possible solutions into one big group. You simply solve the two inequalities completely independently of one another. When graphing them on a number line, you will usually draw two separate arrows pointing in opposite directions. As long as a number falls under at least one of those shaded arrows, it is a valid solution.