Inequalities

⚖️ Inequalities

Part 1 of 5 — Symbols & Meaning

Topics in This Part

🔑 Key Concept: An equation says two things are equal ($x = 5$). An inequality says one thing is bigger or smaller than another ($x > 5$). That small change means an inequality usually has many solutions, not just one.

The Four Inequality Symbols

Each symbol compares a left side to a right side. The "open" end of the symbol always points to the bigger value.

💡 Memory trick: The symbol is like an alligator's mouth — it always opens toward the larger number, because the alligator wants the bigger meal. In $7 > 3$, the mouth opens to the $7$.

The line under $\ge$ and $\le$ means "or equal to" is allowed. So $5 \ge 5$ is true, but $5 > 5$ is false.

Concept Check 🎯

Checking a Solution

A number is a solution of an inequality if it makes the inequality true when you substitute it in.

Example: Is $x = 4$ a solution of $x > 3$?

Replace $x$ with $4$:

$4 > 3 \quad\checkmark \text{ True}$

So $4$ is a solution. But so are $5$, $10$, $3.5$, and $100$ — an inequality has lots of solutions!

Example: Is $x = 3$ a solution of $x > 3$?

$3 > 3 \quad\text{✗ False (3 is not greater than itself)}$

So $3$ is not a solution of $x > 3$. (It would be a solution of $x \ge 3$.)

⚠️ Watch the difference: $x > 3$ does not include $3$ itself. $x \ge 3$ does include $3$. That single line under the symbol changes the answer.

Is It a Solution? 🔽

Substitute each value and decide whether it makes the inequality true.

"At Least," "At Most," and Friends

Real problems rarely use the word inequality. Instead they use everyday phrases. Learn to translate them:

💡 The tricky pair: "at least 18" means $\ge 18$ ($18$ counts!), while "more than 18" means $> 18$ ($18$ does not count). The phrase "at least" always includes the number.

Example: "You must be at least $13$ years old" → $\text{age} \ge 13$. A $13$-year-old qualifies.

Example: "Carry no more than $50$ pounds" → $\text{weight} \le 50$. Exactly $50$ pounds is allowed.

Translate the Words 🎯

⚖️ Inequalities

Part 2 of 5 — Graphing on a Number Line

🔑 The Idea: Because an inequality has many solutions, we don't write them all out — we draw them. A number-line graph shows every solution at once using a dot and an arrow.

Open Dot or Closed Dot?

To graph an inequality, put a dot on the boundary number and shade an arrow toward all the solutions.

And which way does the arrow point?

• Greater ($>$, $\ge$): shade to the right (toward bigger numbers).
• Less ($<$, $\le$): shade to the left (toward smaller numbers).

Example: Graph $x \ge 2$

Put a closed dot on $2$ (because $\ge$ includes $2$) and shade right:

$\xleftarrow{\;\;} \cdots\; 0 \quad 1 \quad \overset{\bullet}{2} \!\!\rightarrow\!\! 3 \quad 4 \;\cdots \xrightarrow{\;\;}$

Example: Graph $x < 2$

Put an open dot on $2$ (because $<$ excludes $2$) and shade left.

💡 Two questions, every time: (1) Open or closed dot? Look for the line under the symbol. (2) Left or right arrow? Greater = right, less = left.

Dot & Direction 🔽

For each inequality, choose the correct dot and arrow direction.

Reading a Graph Back Into an Inequality

You can also go the other way: look at a graph and write its inequality.

1. Find the boundary number (where the dot sits).
2. Open dot → use $>$ or $<$. Closed dot → use $\ge$ or $\le$.
3. Arrow right → "greater" ($>$ or $\ge$). Arrow left → "less" ($<$ or $\le$).

Example

A graph has a closed dot on $-3$ with the arrow shading right.

• Closed dot → "or equal to" → $\ge$
• Arrow right → "greater"

$x \ge -3$

Example

A graph has an open dot on $4$ with the arrow shading left.

• Open dot → strict → $<$
• Arrow left → "less"

$x < 4$

Read the Graph 🎯

Build the Graph 🔽

You are graphing $x \ge -2$. Choose the correct features.

⚖️ Inequalities

Part 3 of 5 — Solving One-Step Inequalities

🔑 Good news: Solving an inequality works almost exactly like solving an equation — you undo operations to get $x$ alone. There is just one new rule, and it's the most important thing in this whole lesson.

Adding & Subtracting

To get $x$ by itself, do the opposite operation to both sides — just like an equation. Adding or subtracting never changes the inequality symbol.

Example: $x + 5 > 12$

Subtract $5$ from both sides:

$x + 5 - 5 > 12 - 5 \;\Rightarrow\; x > 7$

Example: $x - 4 \le 3$

Add $4$ to both sides:

$x - 4 + 4 \le 3 + 4 \;\Rightarrow\; x \le 7$

✅ Check: Pick a number that should work, like $x = 6$ for $x \le 7$. Then $6 - 4 = 2$, and $2 \le 3$ ✓. The solution holds.

Solve by Adding/Subtracting 🧮

Solve each inequality. Enter just the number that $x$ is compared to (the symbol stays the same as the problem).

1) $x + 8 < 15 \;\Rightarrow\; x < \,?$ 2) $x - 6 \ge 1 \;\Rightarrow\; x \ge \,?$ 3) $x + 3 > -2 \;\Rightarrow\; x > \,?$

The One New Rule: Multiplying or Dividing by a Negative

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

$> \;\leftrightarrow\; < \qquad\qquad \ge \;\leftrightarrow\; \le$

Why? Watch a true statement get multiplied by $-1$:

$3 < 5 \quad\text{(true)}$ $-3 \;?\; -5 \quad\Rightarrow\quad -3 > -5 \quad\text{(true)}$

Multiplying by $-1$ swapped which number was bigger, so the symbol had to flip to stay true.

⚠️ The #1 mistake in this whole topic: forgetting to flip. If you multiply or divide by a positive number, the symbol stays the same. Only a negative multiplier or divisor flips it.

Example: $-2x < 8$

Divide both sides by $-2$ and flip $<$ to $>$:

$\frac{-2x}{-2} \;\textbf{>}\; \frac{8}{-2} \;\Rightarrow\; x > -4$

Example: $\dfrac{x}{3} \le 5$

Multiply both sides by $3$ (a positive, so no flip):

$x \le 15$

Do You Flip? 🎯

Putting It Together

Always ask: "Am I multiplying or dividing by a negative?" If yes, flip. If no, keep the symbol.

Example: $-3x \ge -18$

Divide both sides by $-3$ and flip $\ge$ to $\le$:

$x \le 6$

✅ Check: Try $x = 0$ (which satisfies $x \le 6$): $-3(0) = 0$, and $0 \ge -18$ ✓. The solution is correct.

Example: $\dfrac{x}{-2} > 4$

Multiply both sides by $-2$ and flip $>$ to $<$:

$x < -8$

Solve & Watch the Flip 🧮

Solve each. First enter the boundary number, then choose whether the symbol flips.

1) $6x \le 30 \;\Rightarrow\; x \le \,?$ 2) $-4x < 12 \;\Rightarrow\; x \,?\, -3$ (enter the number $-3$ in box 2) 3) $\dfrac{x}{-5} \ge 2 \;\Rightarrow\; x \le \,?$

⚖️ Inequalities

Part 4 of 5 — Two-Step Inequalities & Word Problems

🔑 Big Picture: Most real inequalities take two steps to solve. The order is the reverse of building the expression: undo adding/subtracting first, then undo multiplying/dividing — and remember the flip rule on that last step if you divide by a negative.

Solving in Two Steps

To solve something like $2x + 3 > 11$:

1. Undo the addition/subtraction (the constant) first.
2. Undo the multiplication/division (the coefficient) second.
3. Flip only if step 2 divides by a negative.

Example: $2x + 3 > 11$

$2x + 3 - 3 > 11 - 3 \;\Rightarrow\; 2x > 8$ $\frac{2x}{2} > \frac{8}{2} \;\Rightarrow\; x > 4$

(We divided by $+2$, so no flip.)

Example: $-3x + 1 \le 10$

$-3x + 1 - 1 \le 10 - 1 \;\Rightarrow\; -3x \le 9$

Now divide by $-3$ and flip $\le$ to $\ge$:

$x \ge -3$

✅ Check $x = 0$ (which satisfies $x \ge -3$): $-3(0) + 1 = 1$, and $1 \le 10$ ✓.

Order the Steps 🔽

You're solving $5x - 7 \ge 13$. Choose what happens at each stage.

Two-Step Practice 🧮

Solve each. Enter the boundary number ($x$ is compared to it).

1) $3x + 4 < 19 \;\Rightarrow\; x < \,?$ 2) $\dfrac{x}{2} - 5 \ge 1 \;\Rightarrow\; x \ge \,?$ 3) $-2x + 6 > 14 \;\Rightarrow\; x < \,?$ (remember the flip)

Word Problems With Inequalities

Word problems become inequalities when they include a limit — a budget, a maximum weight, a minimum score. The steps:

1. Decide what the variable stands for.
2. Translate the limit phrase into a symbol ($\le$ "at most", $\ge$ "at least", etc.).
3. Write the inequality and solve.

Example: The Carnival Budget

You have $20. Ride tickets cost $3 each, and you already spent $5 on entry. How many tickets $t$ can you buy?

Total spent must be at most $20:

$3t + 5 \le 20$ $3t \le 15 \;\Rightarrow\; t \le 5$

You can buy at most $5$ tickets.

💡 Reality check: Since $t$ is a count of tickets, it must also be a whole number $\ge 0$. The answer "$t \le 5$" combined with common sense means $0, 1, 2, 3, 4,$ or $5$ tickets.

Word Problem Check 🎯

⚖️ Inequalities

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read and write the four symbols, (2) graph solutions on a number line, (3) solve one- and two-step inequalities, and (4) turn word problems into inequalities. Let's put it all together.

Quick Reference

⚠️ The single most-tested idea: flip the symbol whenever you multiply or divide both sides by a negative number. Nothing else flips it.

Mixed Practice 🎯

Solve, Then Graph 🔽

Solve $-2x + 1 \ge 9$, then describe its graph.

Final Drill 🧮

Solve each and enter the boundary number.

1) $x + 11 > 4 \;\Rightarrow\; x > \,?$ 2) $5x - 2 \le 23 \;\Rightarrow\; x \le \,?$ 3) $\dfrac{x}{-3} > 2 \;\Rightarrow\; x < \,?$ (remember the flip)

Exit Quiz ✅

Answer all three to finish the lesson.