Solving Linear Inequalities

If you have already solved linear equations, you are more than ready for inequalities. Think of inequalities as equations’ more flexible cousin. Instead of asking “what single value makes this true?” inequalities ask “what range of values works?” This is actually more realistic for many real-world situations. After all, how often does a budget constraint require you to spend exactly $50? More likely, you need to spend at most $50 or at least $20. That is where inequalities come in.

Core Concepts

What is an Inequality?

An inequality is a mathematical statement comparing two expressions where one side is not necessarily equal to the other. Instead of the equals sign ($=$), we use inequality symbols to show the relationship.

For example:

• $x < 5$ means “$x$ is less than 5”
• $x > 3$ means “$x$ is greater than 3”
• $x \leq 7$ means “$x$ is less than or equal to 7”
• $x \geq -2$ means “$x$ is greater than or equal to $-2$”

The key difference from equations: while $x + 3 = 7$ has exactly one solution ($x = 4$), the inequality $x + 3 < 7$ has infinitely many solutions. Any number less than 4 works: 3, 2, 0, -100, 3.999, and countless others.

Understanding Inequality Symbols

Here is a helpful way to remember the symbols: the inequality sign always “points to” the smaller value. You can think of it as an alligator mouth that wants to eat the bigger number.

• $3 < 5$ — The “mouth” opens toward 5 because 5 is bigger
• $10 > 2$ — The “mouth” opens toward 10 because 10 is bigger

The symbols $\leq$ and $\geq$ include the possibility of equality. So $x \leq 5$ means $x$ can be 5, or it can be anything smaller than 5.

The symbol $\neq$ simply means “not equal to.” So $x \neq 0$ tells us that $x$ can be any number except zero.

Graphing Inequalities on a Number Line

Since inequalities have infinitely many solutions, we cannot list them all. Instead, we visualize them on a number line.

For $x < 4$:

<-------|-------|-------|---○========>
        1       2       3   4       5

• Open circle at 4 (because 4 itself is not included)
• Shade to the left (all numbers less than 4)

For $x \geq -1$:

<======|---●========|=======>
      -2  -1        0       1

• Closed circle at $-1$ (because $-1$ is included)
• Shade to the right (all numbers $-1$ or greater)

The rule for circles:

• Use an open circle ($\circ$) for $<$ or $>$ — the endpoint is NOT included
• Use a closed circle ($\bullet$) for $\leq$ or $\geq$ — the endpoint IS included

Solving One-Step Inequalities

Here is the good news: solving inequalities works almost exactly like solving equations. You use the same inverse operations to isolate the variable.

Addition/Subtraction: No special rules needed.

If $x + 5 > 12$, subtract 5 from both sides: $$x + 5 - 5 > 12 - 5$$ $$x > 7$$

If $x - 3 \leq 10$, add 3 to both sides: $$x - 3 + 3 \leq 10 + 3$$ $$x \leq 13$$

Multiplication/Division by positive numbers: No special rules needed.

If $4x < 20$, divide both sides by 4: $$\frac{4x}{4} < \frac{20}{4}$$ $$x < 5$$

The Flip Rule: Multiplying or Dividing by Negatives

Here is where inequalities differ from equations, and this is crucial: when you multiply or divide both sides of an inequality by a negative number, you must flip (reverse) the inequality sign.

Why? Consider this: we know $3 < 5$. But if we multiply both sides by $-1$:

• $-1 \times 3 = -3$
• $-1 \times 5 = -5$

Is $-3 < -5$? No! On the number line, $-3$ is to the right of $-5$, so $-3 > -5$.

Multiplying by a negative number reverses the order of numbers. That is why we flip the sign.

Example: Solve $-2x \leq 8$.

Divide both sides by $-2$. Since we are dividing by a negative, flip the sign: $$\frac{-2x}{-2} \geq \frac{8}{-2}$$ $$x \geq -4$$

Common mistake alert: Many students forget to flip the sign when dividing by a negative. Always pause and ask yourself: “Am I multiplying or dividing by a negative number?” If yes, flip the inequality symbol.

Solving Multi-Step Inequalities

Just like multi-step equations, some inequalities require more than one operation. The strategy is the same: use inverse operations in the reverse order (undo addition/subtraction first, then undo multiplication/division).

Example: Solve $3x - 5 < 10$.

Step 1: Add 5 to both sides. $$3x - 5 + 5 < 10 + 5$$ $$3x < 15$$

Step 2: Divide both sides by 3. $$\frac{3x}{3} < \frac{15}{3}$$ $$x < 5$$

The solution is all numbers less than 5.

Compound Inequalities

Sometimes we need to express that a value falls within a certain range. That is where compound inequalities come in. There are two types:

AND compound inequalities (the value must satisfy BOTH conditions):

$-2 < x \leq 5$ means $x$ is greater than $-2$ AND less than or equal to 5.

This can also be written as two separate inequalities:

• $x > -2$ AND $x \leq 5$

On a number line, the solution is the overlap of both conditions:

<---○===========●----->
   -2    0      5

OR compound inequalities (the value must satisfy AT LEAST ONE condition):

$x < -1$ OR $x \geq 3$ means $x$ is either less than $-1$ or greater than or equal to 3.

On a number line, the solution includes BOTH regions:

<====○---|---○===●====>
    -1   0   2   3

Solving compound inequalities:

For AND inequalities written in compact form like $-4 < 2x + 2 \leq 10$, you can solve by performing the same operation on all three parts:

Step 1: Subtract 2 from all three parts. $$-4 - 2 < 2x + 2 - 2 \leq 10 - 2$$ $$-6 < 2x \leq 8$$

Step 2: Divide all three parts by 2. $$\frac{-6}{2} < \frac{2x}{2} \leq \frac{8}{2}$$ $$-3 < x \leq 4$$

For OR inequalities, solve each inequality separately and combine the solutions.

Interval Notation

Mathematicians often use a compact notation called interval notation to describe solution sets.

Key rules:

• Use ( or ) (parentheses) when the endpoint is NOT included
• Use [ or ] (brackets) when the endpoint IS included
• Always use parentheses with $\infty$ and $-\infty$ (infinity is not a number you can reach)
• The symbol $\cup$ means “union” or “OR” — it combines separate intervals

Notation and Terminology

Examples

<-------|-------|-------|---○========>
        2       3       4   5       6

<====|---●========|=======>
    -5  -4       -3      -2

<-------|-------|-------|---○========>
        2       3       4   5       6

<---○===============●--->
   -3  -1   0   2   4   6

<====○---|-------|---○====>
    -2   0       2   3   5

Key Properties and Rules

The Basic Principle

Solving inequalities follows the same logic as solving equations: use inverse operations to isolate the variable. Whatever you do to one side, do to the other.

The Critical Exception: The Flip Rule

When you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality sign.

$$-x > 5 \implies x < -5$$ $$\frac{x}{-2} \leq 3 \implies x \geq -6$$

Compound Inequalities

AND: The solution must satisfy both conditions. The final solution is the overlap (intersection) of both solution sets.

OR: The solution must satisfy at least one condition. The final solution is the combination (union) of both solution sets.

Graphing Guide

Interval Notation Quick Reference

• $($ or $)$ means NOT included
• $[$ or $]$ means included
• Always use $($ $)$ with $\infty$ and $-\infty$
• Use $\cup$ to combine separate intervals (for OR)

Real-World Applications

Budget Constraints

You have a budget of at most $100 for groceries. If you have already spent $35, how much more can you spend?

Let $x$ = additional spending. $$35 + x \leq 100$$ $$x \leq 65$$

You can spend up to $65 more.

Height and Age Requirements

A roller coaster requires riders to be at least 48 inches tall.

If $h$ = height in inches: $$h \geq 48$$

Anyone 48 inches or taller can ride.

Temperature Ranges for Safety

A medication must be stored between 36 and 46 degrees Fahrenheit (inclusive) to remain effective.

If $T$ = temperature: $$36 \leq T \leq 46$$

In interval notation: $[36, 46]$

Grade Boundaries

To pass a class, you need an average of at least 60%. After four tests with scores of 55, 70, 58, and 62, what do you need on the fifth test?

Let $x$ = fifth test score. $$\frac{55 + 70 + 58 + 62 + x}{5} \geq 60$$ $$\frac{245 + x}{5} \geq 60$$ $$245 + x \geq 300$$ $$x \geq 55$$

You need at least 55% on the fifth test to pass.

Speed Limits

A highway has a minimum speed of 45 mph and a maximum of 70 mph.

If $s$ = your speed: $$45 \leq s \leq 70$$

Driving 40 mph or 75 mph would both be violations.

Self-Test Problems

Summary

• An inequality compares values using $<$, $>$, $\leq$, $\geq$, or $\neq$ instead of $=$.
• Unlike equations with one solution, inequalities typically have infinitely many solutions (a range of values).
• Solve inequalities using the same inverse operations as equations: undo addition/subtraction first, then undo multiplication/division.
• The flip rule is critical: when multiplying or dividing by a negative number, reverse the inequality sign.
• Graph inequalities on a number line using open circles ($<$, $>$) or closed circles ($\leq$, $\geq$).
• Compound inequalities combine two conditions with AND (both must be true) or OR (at least one must be true).
• Interval notation provides a compact way to write solution sets using parentheses for excluded endpoints and brackets for included endpoints.
• Always check your solution by testing a value from your solution set in the original inequality.
• Inequalities model many real-world situations involving constraints, requirements, and acceptable ranges.