Lesson 2.6: Solve Compound Inequalities

New Concept

A compound inequality joins two inequalities with 'and' or 'or'. You'll learn to find solutions that satisfy both conditions simultaneously (and) or at least one condition (or), graphing the results and writing them in interval notation.

What’s next

Next, tackle practice cards for 'and' and 'or' inequalities. You'll then apply these skills to interactive examples and real-world challenge problems.

Property

A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve each inequality separately and then consider the two solutions.

Examples

• The statement $x > 2$ and $x < 7$ is a compound inequality. A value like $x=5$ is a solution because it satisfies both conditions.

• The statement $y \leq 0$ or $y \geq 4$ is a compound inequality. A value like $y=6$ is a solution because it satisfies the second condition, even though it fails the first.

Property

To solve a compound inequality with the word “and,” we look for all numbers that make both inequalities true. This is the intersection of the two solution sets. When graphing, the solution is the region where the graphs of the individual inequalities overlap.

Examples

• Solve $4x - 1 < 7$ and $x + 5 \geq 2$. First, we solve each part: $4x < 8$ gives $x < 2$, and $x \geq -3$. The solution is all numbers that are both less than 2 and greater than or equal to -3, which is $[-3, 2)$.

• Solve $2x > 2$ and $x < 5$. This simplifies to $x > 1$ and $x < 5$. The overlapping region is all numbers between 1 and 5, written as $(1, 5)$.

Property

A double inequality is a compound inequality such as $a < x < b$. It is equivalent to $a < x$ and $x < b$. To solve, perform the same operation on all three parts of the inequality to isolate the variable in the middle.

$a \leq x \leq b$ is equivalent to $a \leq x$ and $x \leq b$

$a > x > b$ is equivalent to $a > x$ and $x > b$

Property

To solve a compound inequality with the word “or,” we look for all numbers that make either inequality true. The solution is the union of the two solution sets, meaning it includes all numbers that are in one set, the other set, or both.

Examples

• Solve $3x - 2 > 10$ or $x + 1 < 0$. First, solve each part: $3x > 12$ gives $x > 4$, and $x < -1$. The solution is all numbers less than -1 or greater than 4, written as $(-\infty, -1) \cup (4, \infty)$.

• Solve $x - 5 > 2$ or $x - 5 > 0$. This gives $x > 7$ or $x > 5$. Since any number greater than 7 is also greater than 5, the solutions are combined into the single interval $(5, \infty)$.