Introduction to Inequalities

Equations tell you two things are equal. But in real life, quantities are often not equal — a patient’s temperature must stay below a certain threshold, or an HVAC system must keep a room at least a certain temperature. Inequalities are the mathematical tool for expressing these relationships.

Inequality Symbols

There are four inequality symbols to learn:

Memory aid: The symbol always points to the smaller value. Think of it as an arrow that points at the little number.

The symbols $\geq$ and $\leq$ include the “or equal to” part — the value on the line is also a solution. The symbols $>$ and $<$ are strict — the boundary value is NOT included.

Writing Inequalities from Words

Just like translating words to expressions, certain English phrases map to inequality symbols:

Example 1: “You must be at least 18 years old to vote”

Let $a$ represent age:

$a \geq 18$

Example 2: “The class has fewer than 30 students”

Let $s$ represent the number of students:

$s < 30$

Example 3: “The speed limit is no more than 65 mph”

Let $v$ represent speed:

$v \leq 65$

Graphing Inequalities on a Number Line

A number line graph shows all the values that make an inequality true. Two key conventions:

• Open circle ($\circ$): the endpoint is NOT included (used with $>$ and $<$)
• Closed circle ($\bullet$): the endpoint IS included (used with $\geq$ and $\leq$)

Example 4: Graph $x > 3$

Draw an open circle at 3 and shade everything to the right.

The open circle means 3 itself is not a solution — only numbers strictly greater than 3.

Example 5: Graph $x \leq -1$

Draw a closed circle at $-1$ and shade everything to the left.

The closed circle means $-1$ is included as a solution.

Example 6: Graph $x \geq 0$

Draw a closed circle at 0 and shade to the right. Zero and all positive numbers are solutions.

Solving One-Step Inequalities

Solving inequalities works almost exactly like solving equations — with one critical difference.

Addition and Subtraction: Same as Equations

Example 7: Solve $x + 4 > 9$

Subtract 4 from both sides:

$x > 5$

Graph: Open circle at 5, shade right.

Example 8: Solve $n - 6 \leq 2$

Add 6 to both sides:

$n \leq 8$

Graph: Closed circle at 8, shade left.

Multiplication and Division: The Sign-Flip Rule

When you multiply or divide both sides by a negative number, you must flip the inequality symbol.

Why? Consider: $3 > 1$. Multiply both sides by $-1$: $-3$ and $-1$. On the number line, $-3$ is to the LEFT of $-1$, so $-3 < -1$. The direction reversed.

Example 9: Solve $-2x > 10$

Divide both sides by $-2$ and flip the symbol:

$x < -5$

Check with a test value: Try $x = -6$ (which should satisfy $x < -5$):

$-2(-6) = 12 > 10 \quad \text{— true, so the solution is correct}$

Graph: Open circle at $-5$, shade left.

Example 10: Solve $-3n \leq 12$

Divide both sides by $-3$ and flip:

$n \geq -4$

Check: Try $n = 0$: $-3(0) = 0 \leq 12$ — true.

Graph: Closed circle at $-4$, shade right.

Example 11: Solve $\frac{x}{-4} > 2$

Multiply both sides by $-4$ and flip:

$x < -8$

Check: Try $x = -12$: $\frac{-12}{-4} = 3 > 2$ — true.

When NOT to Flip

If you multiply or divide by a positive number, the symbol stays the same:

Example 12: Solve $5x \geq 35$

Divide both sides by 5 (positive — no flip):

$x \geq 7$

Real-World Application: Nursing — Safe Dosage Range

A medication has a maximum safe dose of 400 mg per day. The patient has already received 150 mg. How much more can be administered?

Let $d$ represent the additional dose:

$150 + d \leq 400$

Subtract 150:

$d \leq 250$

Answer: The patient can receive at most 250 mg more. The inequality (not an equation) is appropriate here because any amount from 0 to 250 mg is acceptable — we need a range, not a single number.

Real-World Application: HVAC — Temperature Requirements

An HVAC technician must keep a server room below 75 degrees Fahrenheit. The current temperature is 68 degrees, and the room gains 2 degrees per hour when the cooling is off. How many hours before the cooling must restart?

Let $h$ represent hours:

$68 + 2h < 75$

Subtract 68:

$2h < 7$

Divide by 2:

$h < 3.5$

Answer: The cooling must restart in fewer than 3.5 hours — after 3.5 hours, the temperature would reach exactly 75 degrees, which violates the “below 75” requirement. In practice, the technician would set the system to restart well before the 3.5-hour mark.

Common Mistakes to Avoid

1. Forgetting to flip the symbol when dividing by a negative. This is the most common inequality error. Remember: multiplying or dividing by a negative reverses the direction.

2. Using the wrong circle on a number line. Open circle for strict inequalities ($>$, $<$); closed circle for “or equal to” ($\geq$, $\leq$).

3. Shading the wrong direction. Greater than shades right; less than shades left. Double-check by testing a value in the shaded region.

4. Flipping when adding or subtracting a negative. The flip rule applies ONLY to multiplication and division. Adding or subtracting a negative does NOT require flipping:

   • $x + (-3) > 5$ becomes $x > 8$ (add 3, no flip)
   • $-2x > 6$ becomes $x < -3$ (divide by $-2$, flip)

5. Confusing “at least” and “at most.” “At least 10” means $x \geq 10$ (10 or more). “At most 10” means $x \leq 10$ (10 or fewer).

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Inequalities compare values using $>$, $<$, $\geq$, and $\leq$
• On a number line: open circle for strict ($>$, $<$), closed circle for inclusive ($\geq$, $\leq$)
• Solve one-step inequalities the same way as equations — with one exception
• Flip the inequality symbol when multiplying or dividing by a negative number
• Do NOT flip when adding or subtracting (even with negatives)
• Always test a value from your solution set to verify the direction is correct
• Inequalities are everywhere in applied fields: dosage limits, temperature ranges, building codes, speed limits, and budget constraints

Return to Pre-Algebra for more topics in this section.