Solving Inequalities

What Is an Inequality? ⚖️

An equation uses an equals sign ($=$) to say that two things are exactly the same.

An inequality is similar, but instead of saying two expressions are equal, it says one is greater than or less than the other.

Here are the four inequality symbols you'll use:

A handy trick: the inequality symbol always opens toward the bigger number. The wide side faces the larger value, and the pointy side faces the smaller one.

One Answer vs. Many Answers 🔢

This is the biggest difference between equations and inequalities.

Equation: $x + 3 = 7$

• This has exactly ONE solution: $x = 4$.

Inequality: $x + 3 > 7$

• This has MANY solutions: $x > 4$.
• Try some numbers: $5$ works, $6$ works, $100$ works, even $4.1$ works!
• There are infinitely many answers.

Key idea: An inequality describes a whole range of numbers, not just a single value. Whenever you solve one, your answer will look like $x > 4$ or $y \le 11$ — a statement about all the numbers that fit.

Solving Works Just Like Equations ✨

Here's the good news: you solve inequalities using the same inverse operations you already use for equations. Whatever you do to one side, you do to the other.

• To undo adding, you subtract.
• To undo subtracting, you add.
• To undo multiplying, you divide.
• To undo dividing, you multiply.

Example: Solve $x + 5 > 12$

Step 1 — Subtract $5$ from both sides: $x + 5 - 5 > 12 - 5$ $x > 7$

So any number greater than 7 is a solution. Let's check $x = 8$: $8 + 5 = 13$, and $13 > 12$ ✓

Quick Concept Check ✅

Let's make sure the big idea is clear.

Worked Example: Add or Subtract ➕➖

When a number is added to or subtracted from the variable, undo it with the opposite operation on both sides.

Example 1 — Solve $y - 3 \le 8$

Step 1 — Add $3$ to both sides (to undo the $-3$): $y - 3 + 3 \le 8 + 3$ $y \le 11$

Answer: $y \le 11$ — any number less than or equal to 11 works.

Check with $y = 11$: $11 - 3 = 8$, and $8 \le 8$ ✓

Worked Example: Multiply or Divide (Positive) ✖️➗

When the variable is multiplied or divided by a positive number, undo it with the opposite operation. Because the number is positive, the symbol stays the same.

Example 2 — Solve $4x < 20$

Step 1 — Divide both sides by $4$: $\frac{4x}{4} < \frac{20}{4}$ $x < 5$

Answer: $x < 5$

Example 3 — Solve $\frac{x}{3} \ge 6$

Step 1 — Multiply both sides by $3$: $3 \times \frac{x}{3} \ge 3 \times 6$ $x \ge 18$

Answer: $x \ge 18$

Your Turn: Solve These ✍️

Solve each one-step inequality. Type only the number that goes after the symbol (for example, if the answer is $x > 7$, just type 7).

1. Solve $x + 6 > 10$. Then $x > \;?$
2. Solve $3x < 21$. Then $x < \;?$
3. Solve $\frac{y}{2} \ge 5$. Then $y \ge \;?$

⚠️ The Special Rule: Flip for Negatives!

Here's the one rule that makes inequalities different from equations:

When you multiply or divide both sides by a NEGATIVE number, you must FLIP the inequality symbol!

Why? Start with a true statement: $5 > 2$.

• Multiply both sides by $-1$: we get $-5$ and $-2$.
• On a number line, $-5$ sits to the left of $-2$, so $-5 < -2$.
• The symbol flipped from $>$ to $<$ to keep the statement true!

Example — Solve $-3x > 12$

Step 1 — Divide both sides by $-3$ and flip $>$ to $<$: $\frac{-3x}{-3} < \frac{12}{-3}$ $x < -4$

Answer: $x < -4$. Check $x = -5$: $-3(-5) = 15$, and $15 > 12$ ✓

Now try the questions below.

Multiple Choice 🎯

Flip or Don't Flip? 🔽

For each problem, choose whether you flip the symbol, and pick the correct solution.

Inequalities in the Real World 🌎

Inequalities are everywhere — anytime we talk about a limit, a minimum, or a maximum.

Watch for these clue words:

• "at most" or "no more than" → $\le$
• "at least" or "no less than" → $\ge$
• "less than" or "under" → $<$
• "more than" or "over" → $>$

Example: A ride at the fair has a sign: "You must be at least 48 inches tall."

If $h$ stands for your height, this becomes the inequality $h \ge 48$. Any height of 48 inches or more lets you ride.

Example with a solve: Tickets cost $8 each, and Jordan has $50 to spend. How many tickets $t$ can Jordan buy? $8t \le 50$ $t \le 6.25$ Since Jordan can't buy part of a ticket, the most they can buy is 6 tickets.

Set Up and Solve ✍️

Type only the number that goes after the inequality symbol.

1. A bus holds at most 40 people. Write $p \le \;?$ for the number of people allowed.
2. To pass, Maria needs at least an 80 on her test. Write $s \ge \;?$ for her score $s$.
3. A pizza costs $9 and Sam has $45. Solving $9p \le 45$ gives $p \le \;?$ pizzas.

Word Problem Check 🎯

Putting It All Together 🧠

You now know how to solve and graph inequalities. Here's everything in one place:

Remember the symbols:

• $<$ less than, $>$ greater than
• $\le$ less than or equal to, $\ge$ greater than or equal to

The #1 thing to watch: only flip the inequality symbol when you multiply or divide by a negative number. Adding and subtracting never flip it!

Ready for the final challenge? 💪

Final Challenge 🏆

These mix everything together. Take your time on each one.