2.7 Solve Linear Inequalities

Linear Inequalities

Linear inequalities work like equations, but instead of finding one exact answer, you find a range of values that make the statement true. They use symbols like $<$, $>$, $\leq$, and $\geq$ to compare expressions. You solve them almost the same way you solve equations, with one critical exception: multiplying or dividing by a negative number flips the inequality symbol.

Linear Inequalities on Number Lines

The four inequality symbols each mean something specific:

• $<$ means "less than"
• $>$ means "greater than"
• $\leq$ means "less than or equal to"
• $\geq$ means "greater than or equal to"

To graph an inequality on a number line, you need two things: the right type of circle and the right direction of shading.

• Open circle (○) for strict inequalities ($<$ or $>$), meaning the number itself is not included
• Closed circle (●) for inclusive inequalities ($\leq$ or $\geq$), meaning the number is included
• Shade left when the variable is less than the value
• Shade right when the variable is greater than the value

For example, $x < 5$ gets an open circle at 5 with shading to the left. The inequality $x \geq -2$ gets a closed circle at $-2$ with shading to the right.

Addition and Subtraction in Inequalities

Adding or subtracting the same value on both sides of an inequality does not change the direction of the symbol. This works exactly like solving an equation.

Example: Solve $x - 3 < 7$

1. Add 3 to both sides: $x - 3 + 3 < 7 + 3$

2. Simplify: $x < 10$

The solution means any value less than 10 makes the original inequality true. On a number line, you'd place an open circle at 10 and shade to the left.

Multiplication and Division in Inequalities

Multiplying or dividing both sides by a positive number works just like with equations. The inequality symbol stays the same.

Multiplying or dividing both sides by a negative number requires you to flip the inequality symbol. This is the single biggest difference between solving equations and solving inequalities.

Why does the symbol flip? Think about it with simple numbers: $2 < 5$ is true. But if you multiply both sides by $-1$, you get $-2$ and $-5$. Since $-2$ is actually greater than $-5$, the relationship reverses to $-2 > -5$.

Example: Solve $-2x > 6$

1. Divide both sides by $-2$ (a negative number, so flip the symbol): $\frac{-2x}{-2} < \frac{6}{-2}$
2. Simplify: $x < -3$

Notice the $>$ became $<$ because you divided by a negative.

Complex Linear Inequalities

When an inequality has variables and constants on both sides, combine like terms first, then isolate the variable step by step.

Example: Solve $3x - 5 + 2x \geq 4x - 7$

1. Combine like terms on the left side: $5x - 5 \geq 4x - 7$

2. Subtract $4x$ from both sides: $x - 5 \geq -7$

3. Add 5 to both sides: $x \geq -2$

The solution is all values greater than or equal to $-2$. On a number line, place a closed circle at $-2$ and shade to the right.

Word Problems to Inequalities

Translating word problems into inequalities requires matching key phrases to the correct symbol:

• "at most" or "no more than" → $\leq$
• "at least" or "no fewer than" → $\geq$
• "fewer than" or "less than" → $<$
• "more than" or "greater than" → $>$

Example: The sum of three times a number and 5 is at most 20. Find the range of possible values.

1. Let $x$ represent the unknown number
2. Translate: "three times a number" is $3x$; "sum of ... and 5" gives $3x + 5$; "is at most 20" means $\leq 20$
3. Write the inequality: $3x + 5 \leq 20$
4. Subtract 5 from both sides: $3x \leq 15$
5. Divide both sides by 3: $x \leq 5$

The number can be any value less than or equal to 5.

Graphing Solutions

After solving an inequality, graphing the solution on a number line helps you visualize which values work. Every solution you find in this section can be represented as a ray on a number line, starting at the boundary value and extending in one direction. Always check whether the boundary point is included (closed circle) or excluded (open circle) based on the inequality symbol.