Linear Inequalities

⚖️ Linear Inequalities

Part 1 of 5 — Reading and Understanding Inequalities

Topics in This Part

🔑 Key Concept: An equation like $x = 3$ has one answer. An inequality like $x > 3$ has infinitely many answers — every number bigger than $3$. Learning to read, solve, and picture those ranges is what this lesson is about.

The Four Inequality Symbols

Each symbol compares two quantities and points (the small end) at the smaller value.

💡 Strict vs. inclusive: $<$ and $>$ are strict — they exclude the boundary number. $\le$ and $\ge$ are inclusive — they include it. That single difference decides whether a graph uses an open circle or a closed dot (Part 4).

You can also flip a statement around as long as you flip the symbol with it:

$x > 5 \quad\text{means the same as}\quad 5 < x$

Concept Check 🎯

Solutions Are Ranges

The solution of an inequality is the set of all numbers that make it true.

Take $x > 3$. Is $4$ a solution? Yes ($4 > 3$). Is $3.001$? Yes. Is $100$? Yes. Is $3$ itself? No — the symbol is strict. So the solution is every number greater than $3$ — an infinite range.

🔑 The big shift from equations: When you "solve" an inequality, you're not hunting for one value — you're describing a whole interval of values. The answer is itself an inequality, like $x \le 8$.

Words That Become Symbols

Translate the Words 🔽

Pick the inequality that matches each phrase (let $x$ be the number).

Checking a Candidate

To test whether a number is a solution, substitute it and see if the statement is true.

Example: Is $x = 2$ a solution of $3x - 1 \le 5$?

$3(2) - 1 = 6 - 1 = 5 \le 5 \;\checkmark$

Yes — $5 \le 5$ is true (the $\le$ allows equality), so $2$ is a solution.

Example: Is $x = -1$ a solution of $2x + 4 > 3$?

$2(-1) + 4 = -2 + 4 = 2 > 3 \;?$

$2 > 3$ is false, so $-1$ is not a solution.

💡 Checking is your safety net. After you solve an inequality in Part 2, plug a number from your answer range back in to confirm it works.

Concept Check 🎯

⚖️ Linear Inequalities

Part 2 of 5 — Solving One- and Two-Step Inequalities

🔑 The Big Idea: You solve an inequality almost exactly like an equation — add, subtract, multiply, and divide to isolate the variable. There is one new rule, and it is the most important rule in the whole lesson.

Most Steps Work Just Like Equations

Adding or subtracting the same number from both sides keeps an inequality true and keeps the same symbol.

Example: $x + 5 < 9$

$x + 5 - 5 < 9 - 5 \;\Rightarrow\; x < 4$

Example: $x - 3 \ge 2$

$x - 3 + 3 \ge 2 + 3 \;\Rightarrow\; x \ge 5$

Multiplying or dividing by a positive number also keeps the symbol the same.

Example: $4x \le 20$

$\frac{4x}{4} \le \frac{20}{4} \;\Rightarrow\; x \le 5$

✅ Check $x = 5$: $4(5) = 20 \le 20$ ✓, and a smaller value like $x=0$ gives $0 \le 20$ ✓.

Same as an Equation? 🔽

For each step, choose whether the inequality symbol stays the same or flips.

⚠️ The Flip Rule (the one new rule)

⚠️ When you multiply or divide BOTH sides by a NEGATIVE number, you MUST reverse the inequality symbol.

Why? Watch what negatives do to order. We know $2 < 6$. Multiply both sides by $-1$:

$-2 \quad\text{vs.}\quad -6$

Now $-2$ is the bigger number, so the true statement is $-2 > -6$. The symbol had to flip.

Example: $-3x < 12$

Divide both sides by $-3$ — and flip $<$ to $>$:

$\frac{-3x}{-3} > \frac{12}{-3} \;\Rightarrow\; x > -4$

Example: $-x \ge 7$

Multiply both sides by $-1$ — flip $\ge$ to $\le$:

$x \le -7$

💡 The flip happens only for multiplying/dividing by a negative. Adding or subtracting a negative number does not flip anything.

Concept Check 🎯

Two-Step Inequalities

Undo the operations in reverse order: deal with addition/subtraction first, then multiplication/division.

Example: $2x + 3 \le 11$

$2x + 3 - 3 \le 11 - 3 \;\Rightarrow\; 2x \le 8 \;\Rightarrow\; x \le 4$

(Divided by a positive $2$ — no flip.)

Example: $-5x + 1 > 16$

$-5x > 15 \;\Rightarrow\; x < \frac{15}{-5} \;\Rightarrow\; x < -3$

(Divided by a negative $-5$ — flipped $>$ to $<$.)

✅ Check $x = -4$ in $-5x + 1 > 16$: $-5(-4)+1 = 21 > 16$ ✓.

Solve It 🧮

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

1) $x + 7 < 2 \;\Rightarrow\; x < \,?$ 2) $3x \ge -12 \;\Rightarrow\; x \ge \,?$ 3) $-4x \le 20 \;\Rightarrow\; x \ge \,?$ (remember to flip)

⚖️ Linear Inequalities

Part 3 of 5 — Multi-Step Inequalities

🔑 Same playbook, more moves: Distribute, combine like terms, and collect the variable on one side — exactly as you would for an equation. Keep watching for the one place a negative can sneak in and flip your symbol.

Distributing First

When a number multiplies a group, distribute before isolating.

Example: $3(x - 2) \le 9$

$3x - 6 \le 9 \;\Rightarrow\; 3x \le 15 \;\Rightarrow\; x \le 5$

Example: $-2(x + 1) > 8$

$-2x - 2 > 8 \;\Rightarrow\; -2x > 10 \;\Rightarrow\; x < -5$

⚠️ Notice the flip in the second example: the final divide was by $-2$, so $>$ became $<$. Distributing the $-2$ at the start did not flip anything — only the divide-by-negative step does.

Concept Check 🎯

Variables on Both Sides

Move all variable terms to one side and all constants to the other. A clean trick: move the variable to whichever side keeps its coefficient positive, and you'll often avoid the flip entirely.

Example: $5x - 4 > 2x + 11$

Subtract $2x$ from both sides, then add $4$:

$3x - 4 > 11 \;\Rightarrow\; 3x > 15 \;\Rightarrow\; x > 5$

Example: $x + 8 \le 4x - 1$

Subtract $x$ from both sides, then add $1$:

$8 \le 3x - 1 \;\Rightarrow\; 9 \le 3x \;\Rightarrow\; 3 \le x \;\Rightarrow\; x \ge 3$

💡 By moving the $x$-terms to the side that kept a positive coefficient ($3x$), neither example needed a flip. Same answer either way — this route is just safer.

Order the Steps 🔽

You're solving $-3(x - 1) \ge 12$. Choose what happens at each stage.

Two Special Outcomes

Sometimes the variable disappears. What's left tells you the answer.

All real numbers (always true)

$2(x + 3) > 2x + 1 \;\Rightarrow\; 2x + 6 > 2x + 1 \;\Rightarrow\; 6 > 1$

The $x$ cancels and $6 > 1$ is always true, so every number works — the solution is all real numbers.

No solution (never true)

$x + 5 < x + 2 \;\Rightarrow\; 5 < 2$

The $x$ cancels and $5 < 2$ is never true, so no number works — there is no solution.

💡 When the variable vanishes, look at the leftover number statement: if it's true, all reals are solutions; if it's false, there are no solutions.

Solve It 🧮

Solve and enter the boundary number ($x$ is compared to it).

1) $4(x + 2) \ge 4 \;\Rightarrow\; x \ge \,?$ 2) $6x - 5 < 2x + 7 \;\Rightarrow\; x < \,?$ 3) $2 - x > 9 \;\Rightarrow\; x < \,?$ (watch the flip)

⚖️ Linear Inequalities

Part 4 of 5 — Graphing, Interval Notation & Word Problems

🔑 See it and use it: A solution like $x > 3$ is easy to picture on a number line and to write compactly in interval notation. Then we'll turn real-world sentences into inequalities and solve them.

Graphing on a Number Line

Two decisions: the type of dot and the direction of the arrow.

Example: $x \le 2$

Closed dot ● on $2$ (because of the $=$ part), arrow shading left toward smaller numbers.

Example: $x > -1$

Open circle ○ on $-1$ (strict, so $-1$ is excluded), arrow shading right toward larger numbers.

⚠️ Open vs. closed is decided by the symbol, not the sign of the number. $x > -1$ uses an open circle because $>$ is strict — the negative has nothing to do with it.

Match the Graph 🔽

Describe the number-line graph of each solution.

Interval Notation

A compact way to write a solution range. Use a square bracket $[$ or $]$ when the endpoint is included ($\le, \ge$) and a parenthesis $($ or $)$ when it is excluded ($<, >$). Infinity $\infty$ always gets a parenthesis (you can never "reach" it).

💡 Bracket = included, parenthesis = excluded. The closed dot ● matches a bracket; the open circle ○ matches a parenthesis. Infinity is never reached, so it is always a parenthesis.

Concept Check 🎯

Word Problems

Translate, solve, then interpret the answer in context.

Example: Budget

A taxi charges a $3 base fare plus $2 per mile. You have $15. How many miles $m$ can you ride?

$3 + 2m \le 15 \;\Rightarrow\; 2m \le 12 \;\Rightarrow\; m \le 6$

You can ride at most $6$ miles.

Example: Test Score

You scored $78$, $85$, and $90$ on three tests. To average at least $85$ over four tests, what score $s$ do you need on the fourth?

$\frac{78 + 85 + 90 + s}{4} \ge 85 \;\Rightarrow\; \frac{253 + s}{4} \ge 85$

$253 + s \ge 340 \;\Rightarrow\; s \ge 87$

You need at least $87$.

💡 "At most" and "no more than" become $\le$; "at least" and "minimum" become $\ge$. Re-read the question to make sure your boundary answer makes sense.

Apply It 🧮

1) A gym charges a $20 sign-up fee plus $10 per month. With a $90 budget, the most months $m$ you can afford satisfies $20 + 10m \le 90$. Solve: $m \le \,?$

2) A rideshare charges $4 plus $1.50 per mile. With $25, solve $4 + 1.5m \le 25$ for the most miles: $m \le \,?$

3) You have $40$ and $52$ on two quizzes. To average at least $50$, solve $\dfrac{40 + 52 + s}{3} \ge 50$ for the third quiz: $s \ge \,?$

⚖️ Linear Inequalities

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read and translate inequalities, (2) solve them with the flip rule, (3) handle multi-step problems, and (4) graph, write intervals, and tackle word problems. Time to put it all together.

Quick Reference

⚠️ The one mistake that ruins an otherwise perfect solution is forgetting to flip the symbol when you multiply or divide by a negative. Circle every negative divisor as a reminder.

Mixed Practice 🎯

Mixed Practice 🔽

Finish each statement.

Exit Quiz ✅

Answer all three to finish the lesson.