Lesson 10: Solving and Graphing Inequalities

New Concept

An inequality is a mathematical statement comparing quantities that are not equal. There are four inequality symbols: $<, >, \leq, \geq$.

What’s next

Next, you’ll master the rules for solving these inequalities and learn how to graph their solutions on a number line.

Property

If $c < 0$ and $a < b$, then $ac > bc$ and $\frac{a}{c} > \frac{b}{c}$.
If $c > 0$ and $a < b$, then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$.

Solve $-4x > 20$. Dividing by $-4$ flips the sign: $\frac{-4x}{-4} < \frac{20}{-4}$, so $x < -5$.
Solve $\frac{y}{-3} \leq 2$. Multiplying by $-3$ flips the sign: $(\frac{y}{-3}) \cdot (-3) \geq 2 \cdot (-3)$, so $y \geq -6$.
Solve $6a < 18$. Dividing by a positive 6 means no flip is needed: $\frac{6a}{6} < \frac{18}{6}$, so $a < 3$.

Think of an inequality like a balanced seesaw. Multiplying by a positive number keeps it level. But multiplying or dividing by a negative number is like swapping the kids to opposite ends—the seesaw flips! To keep the comparison true, you must flip the inequality sign. It's the one tricky rule you must always remember to check for.

Property

To graph an inequality on a number line: use an open circle for $<$ or $>$ to show the number is not included. Use a closed (solid) circle for $\leq$ or $\geq$ to show the number is included in the solution.

The graph of $x > 4$ has an open circle at 4 with an arrow pointing to the right, towards infinity.
The graph of $y \leq -2$ has a closed, solid circle at -2 with an arrow pointing to the left.

Graphing makes the solution visible. Imagine the circle is a gate. An open circle means the value is the starting point, but not part of the yard—you can get close, but not touch it. A closed circle means the value is the gatepost itself, firmly planted in the solution yard. The arrow shows which way the yard extends.

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!