Solving and Graphing Inequalities

⚖️ Solving and Graphing Inequalities

Part 1 of 5 — What Inequalities Mean

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 describe and draw that whole set of answers is what this lesson is about.

The Four Inequality Symbols

An inequality compares two amounts that are not (necessarily) equal. There are four symbols you must know cold:

💡 Memory trick: The symbol is like a hungry mouth — it always opens toward the bigger number. In $7 > 2$, the wide opening faces the $7$.

The line under $\le$ and $\ge$ adds the words "or equal to." That single line decides whether the boundary number is included in the answer — a detail that matters a lot when we graph.

Concept Check 🎯

Solutions Are Whole Sets of Numbers

The solution of an inequality is every number that makes it true.

Take $x > 3$. Is $4$ a solution? Yes — $4 > 3$. Is $3.5$? Yes. Is $1000$? Yes. Is $3$ itself? No, because $3 > 3$ is false ($3$ is not greater than $3$).

$x > 3 \quad\Longrightarrow\quad \text{solutions: } 3.01,\ 4,\ 5,\ 100,\ \dots \ (\text{everything past } 3)$

⚠️ Common confusion: $x > 3$ does not include $3$. But $x \ge 3$ does include $3$. The little line under the symbol is the only difference — and it changes the answer set.

Quick test: plug a number in

To check whether a value is a solution, substitute it and see if the statement is true.

Is It a Solution? 🎯

Translate the Words 🔽

Real problems hide inequalities inside everyday phrases. Pick the symbol that matches each phrase, using $x$ for the unknown amount.

Part 1 Recap

• Four symbols: $<,\ >,\ \le,\ \ge$. The mouth opens toward the bigger value.
• $\le$ and $\ge$ include the boundary number; $<$ and $>$ exclude it.
• "At least / no more than" include the boundary; "more than / less than" exclude it.
• A solution is the entire set of numbers that make the inequality true — test a value by substituting it in.

🔑 Next up (Part 2): drawing these solution sets on a number line, where the include/exclude distinction becomes an open or closed dot.

⚖️ Solving and Graphing Inequalities

Part 2 of 5 — Graphing on a Number Line

🔑 The Idea: A picture beats a list of infinitely many numbers. We draw the solution set as a dot (open or closed) on the boundary plus an arrow showing every number that works.

Open Dot vs. Closed Dot

Two rules cover every single-variable inequality graph:

Rule 1 — the dot:

• Use an open circle $\circ$ for $<$ and $>$ → the boundary is NOT included.
• Use a closed (filled) circle $\bullet$ for $\le$ and $\ge$ → the boundary IS included.

Rule 2 — the arrow:

• Shade/arrow to the right for $>$ and $\ge$ (greater = bigger = right).
• Shade/arrow to the left for $<$ and $\le$ (less = smaller = left).

💡 Tie the dot to the line: the "or equal to" line under $\le$ / $\ge$ becomes the filled-in dot. No line → no fill → open dot.

Reading a Graph

Here is the graph of $x \ge -1$:

        ●━━━━━━━━━▶
 ──┼──┼──┼──┼──┼──┼──┼──
  -3 -2 -1  0  1  2  3

The filled dot sits on $-1$ (so $-1$ is included), and the arrow runs right through all larger numbers. That matches $x \ge -1$.

Now the graph of $x < 2$:

 ◀━━━━━━━━━○
 ──┼──┼──┼──┼──┼──┼──┼──
  -1  0  1  2  3  4  5

The open dot on $2$ (so $2$ is excluded), arrow running left. That is $x < 2$.

⚠️ Don't mix them up: an open dot with a left arrow is $x < (\text{number})$, not $x \le$. The fill is what tells you whether to use the "or equal to" line.

Match the Graph to the Inequality 🔽

For each description of a number-line graph, pick the matching inequality.

Choose the Correct Graph 🎯

Identify the Boundary 🧮

For each inequality, enter the boundary number (where the dot sits on the number line).

1) $x < 7$ → boundary $= \,?$ 2) $x \ge -4$ → boundary $= \,?$ 3) $x \le 0$ → boundary $= \,?$

⚖️ Solving and Graphing Inequalities

Part 3 of 5 — Solving with Add & Subtract

🔑 Great news: Solving an inequality works almost exactly like solving an equation. You undo operations to get the variable alone. Adding or subtracting the same amount on both sides never changes the inequality symbol.

The Addition / Subtraction Property

You may add or subtract the same number on both sides of an inequality, and the symbol stays the same.

$\text{If } a > b, \text{ then } a + c > b + c \quad\text{and}\quad a - c > b - c$

Worked Example: $x - 4 > 3$

Add $4$ to both sides to isolate $x$:

$x - 4 > 3$ $x - 4 + 4 > 3 + 4$ $x > 7$

✅ Check: try $x = 10$ (which is $> 7$): $10 - 4 = 6 > 3$ ✓. Try $x = 7$: $7 - 4 = 3$, and $3 > 3$ is false — correct, since $7$ is not in $x > 7$.

Worked Example: $x + 5 \le 2$

Subtract $5$ from both sides:

$x + 5 \le 2$ $x + 5 - 5 \le 2 - 5$ $x \le -3$

The symbol $\le$ is unchanged. Graph: closed dot on $-3$, arrow left.

Worked Example: $-6 < x - 1$

Add $1$ to both sides:

$-6 + 1 < x - 1 + 1 \;\Rightarrow\; -5 < x$

This reads "$-5$ is less than $x$," which is the same as $x > -5$.

💡 Flip to read it easily: $-5 < x$ and $x > -5$ say the exact same thing — just swap the sides and reverse the symbol so the variable is on the left.

Concept Check 🎯

Solve It 🧮

Solve each inequality. Enter the boundary number for $x$ (just the number — the symbol does not change here).

1) $x + 6 < 10 \;\Rightarrow\; x < \,?$ 2) $x - 3 \ge -8 \;\Rightarrow\; x \ge \,?$ 3) $x + 12 \le 5 \;\Rightarrow\; x \le \,?$

Solve, Then Graph 🔽

Solve each inequality, then describe its graph.

⚖️ Solving and Graphing Inequalities

Part 4 of 5 — Multiply, Divide & The Flip Rule

⚠️ The one big difference from equations: when you multiply or divide both sides by a negative number, you MUST flip the inequality symbol. This is the single most important rule in the whole lesson.

Multiplying or Dividing by a POSITIVE Number

No surprises here — the symbol stays the same, just like with addition.

Worked Example: $3x \le 12$

Divide both sides by $3$ (positive):

$\frac{3x}{3} \le \frac{12}{3} \;\Rightarrow\; x \le 4$

Worked Example: $\dfrac{x}{2} > 5$

Multiply both sides by $2$ (positive):

$2 \cdot \frac{x}{2} > 2 \cdot 5 \;\Rightarrow\; x > 10$

💡 As long as the number you multiply or divide by is positive, the inequality behaves exactly like an equation.

Multiplying or Dividing by a NEGATIVE Number — FLIP!

Here is why the symbol must flip. Start with a true statement:

$3 < 5 \quad (\text{true})$

Multiply both sides by $-1$. If we kept the symbol we'd get $-3 < -5$, which is false ($-3$ is actually bigger than $-5$). To stay true, we must flip the symbol:

$3 < 5 \;\xrightarrow{\;\times(-1)\;}\; -3 > -5 \quad (\text{true!})$

Worked Example: $-2x < 8$

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

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

Worked Example: $-x \ge 6$

Multiply (or divide) both sides by $-1$ and flip $\ge$ to $\le$:

$x \le -6$

⚠️ Flip only for a NEGATIVE multiplier/divisor. Adding or subtracting a negative does not flip — only multiplying or dividing by a negative does.

Flip or Don't Flip? 🔽

For each step, decide whether the inequality symbol flips.

Solve with the Flip Rule 🎯

Solve It 🧮

Solve each. Enter the boundary number for $x$. Watch for the flip!

1) $5x \ge 35 \;\Rightarrow\; x \ge \,?$ 2) $-3x < 18 \;\Rightarrow\; x > \,?$ (flipped to $>$) 3) $\dfrac{x}{-4} \ge 2 \;\Rightarrow\; x \le \,?$ (flipped to $\le$)

⚖️ Solving and Graphing Inequalities

Part 5 of 5 — Two-Step Inequalities & Mastery Check

You can now (1) read inequality symbols, (2) graph them on a number line, (3) solve with $+$ and $-$, and (4) solve with $\times$ and $\div$ (including the flip). Now we combine the steps and finish with an Exit Quiz.

Two-Step Inequalities

Same order of operations as a two-step equation: undo addition/subtraction first, then undo multiplication/division — and flip the symbol only if that last division is by a negative.

Worked Example: $2x + 3 < 11$

$2x + 3 < 11$ $2x < 8 \quad (\text{subtract } 3)$ $x < 4 \quad (\text{divide by } +2,\ \text{no flip})$

Worked Example: $-3x + 1 \ge 10$

$-3x + 1 \ge 10$ $-3x \ge 9 \quad (\text{subtract } 1)$ $x \le -3 \quad (\text{divide by } -3,\ \textbf{flip}\ \ge \to \le)$

⚠️ The flip only happens at the divide-by-negative step. Subtracting the $1$ first did not flip anything.

Quick Reference

🔑 The flip rule is the one thing that makes inequalities different from equations. Master it and you have mastered the topic.

Mixed Practice 🎯

Two-Step Solve 🧮

Solve each. Enter the boundary number for $x$.

1) $3x + 2 \le 17 \;\Rightarrow\; x \le \,?$ 2) $\dfrac{x}{2} - 4 > 1 \;\Rightarrow\; x > \,?$ 3) $-5x - 3 \ge 12 \;\Rightarrow\; x \le \,?$ (remember the flip)

Exit Quiz ✅

Answer all three to finish the lesson.