Solving and Graphing Inequalities

An **inequality** is a mathematical statement that compares two expressions using inequality symbols: | Symbol | Meaning | Example | |--------|---------|--------| | $<$ | Less than | $x < 5$ | | $>$ | Greater than | $x > 3$ | | $\leq$ | Less than or equal to | $x \leq 7$ | | $\geq$ | Greater than or equal to | $x \geq -2$ | Unlike equations (which have one solution), inequalities have **infinitely many solutions**. For example, $x > 3$ means $x$ can be 4, 5, 10, 3.5, or any number greater than 3. **Key Rule**: When you multiply or divide both sides of an inequality by a **negative number**, you must **flip the inequality sign**. $$-2x > 6 \implies x < -3$$

One-Step Inequality (Addition)

Solve $x + 5 > 12$ and graph the solution.

Identify the operation: 5 is being added to $x$ = Need to subtract 5

Subtract 5 from both sides: $x + 5 - 5 > 12 - 5$ = $x > 7$

Graph on number line: Open circle at 7, shade to the right = All numbers greater than 7

Answer: $x > 7$ (open circle at 7, arrow pointing right)

One-Step Inequality (Multiplication)

Solve $3x \leq 18$ and graph the solution.

Identify the operation: $x$ is being multiplied by 3 = Need to divide by 3

Divide both sides by 3: $\frac{3x}{3} \leq \frac{18}{3}$ = $x \leq 6$

Check: Is sign flip needed?: Dividing by positive 3, no flip needed = Sign stays $\leq$

Graph on number line: Closed circle at 6, shade to the left = All numbers 6 or less

Answer: $x \leq 6$ (closed circle at 6, arrow pointing left)

Inequality with Negative Coefficient

Solve $-4x > 20$ and graph the solution.

Identify the operation: $x$ is being multiplied by $-4$ = Need to divide by $-4$

Divide both sides by $-4$: $\frac{-4x}{-4} > \frac{20}{-4}$ = $x \text{ ? } -5$

FLIP the inequality sign!: Dividing by negative number flips the sign = $x < -5$

Graph on number line: Open circle at $-5$, shade to the left = All numbers less than $-5$

Answer: $x < -5$ (open circle at $-5$, arrow pointing left)

Two-Step Inequality

Solve $2x + 7 \geq 15$ and graph the solution.

Subtract 7 from both sides: $2x + 7 - 7 \geq 15 - 7$ = $2x \geq 8$

Divide both sides by 2: $\frac{2x}{2} \geq \frac{8}{2}$ = $x \geq 4$

Check: Is sign flip needed?: Dividing by positive 2, no flip needed = Sign stays $\geq$

Graph on number line: Closed circle at 4, shade to the right = All numbers 4 or greater

Answer: $x \geq 4$ (closed circle at 4, arrow pointing right)

Two-Step with Negative Coefficient

Solve $5 - 3x < 14$ and graph the solution.

Subtract 5 from both sides: $5 - 3x - 5 < 14 - 5$ = $-3x < 9$

Divide both sides by $-3$: $\frac{-3x}{-3} < \frac{9}{-3}$ = $x \text{ ? } -3$

FLIP the inequality sign!: Dividing by $-3$ (negative) flips the sign = $x > -3$

Graph on number line: Open circle at $-3$, shade to the right = All numbers greater than $-3$

Answer: $x > -3$ (open circle at $-3$, arrow pointing right)

Inequalities describe real-world situations with ranges and limits: - **Age requirements**: You must be at least 16 to drive ($\text{age} \geq 16$) - **Speed limits**: Drive no faster than 50 km/h ($\text{speed} \leq 50$) - **Budget constraints**: Spend less than 100 dollars ($\text{cost} < 100$) - **Temperature ranges**: Water stays liquid between 0 and 100 degrees Celsius - **Grades**: You need at least 60 points to pass ($\text{score} \geq 60$) Inequalities help us express conditions, constraints, and ranges that appear everywhere in daily life!

Mistake: Forgetting to flip the sign when dividing by a negative

Why: The most common error! When dividing (or multiplying) by a negative number, the inequality direction reverses.

Correct: $-2x > 8$ becomes $x < -4$ (not $x > -4$). The sign MUST flip.

Mistake: Using closed circle for strict inequalities ($<$ or $>$)

Why: Strict inequalities ($<$, $>$) do NOT include the boundary value.

Correct: $x > 5$: Open circle at 5 (5 is not included). $x \geq 5$: Closed circle at 5 (5 is included).

Mistake: Shading in the wrong direction

Why: Students sometimes shade based on the sign without checking the variable's position.

Correct: Always test a value: For $x > 3$, test $x = 5$. Since $5 > 3$ is true, shade the region containing 5.

Mistake: Flipping the sign when subtracting negatives

Why: The flip rule ONLY applies to multiplication and division by negatives, not addition/subtraction.

Correct: $x - (-5) > 2$ becomes $x + 5 > 2$, then $x > -3$. No flip needed for subtraction.

Budget Planning

Determine how many items you can buy while staying within a budget.

Concert tickets cost 25 dollars each, plus a 15 dollar service fee. If you have 90 dollars, how many tickets can you buy? $25t + 15 \leq 90$

Speed and Distance

Calculate safe driving speeds to arrive on time.

You need to drive at least 200 km in 4 hours. What average speed is required? $4s \geq 200$, so $s \geq 50$ km/h.

Inequalities compare expressions using $<$, $>$, $\leq$, $\geq$

Solve inequalities like equations: add, subtract, multiply, divide on both sides

**Critical rule**: Flip the inequality sign when multiplying or dividing by a negative number

Graph solutions: open circle for $<$ and $>$, closed circle for $\leq$ and $\geq$

Shade the number line in the direction of the solution set

Q: When do I flip the inequality sign?

A: Only when multiplying or dividing both sides by a **negative** number. Adding or subtracting (even negative numbers) does not require flipping.

Q: What is the difference between < and \leq?

A: $<$ means strictly less than (not equal), shown with an open circle. $\leq$ means less than OR equal to, shown with a closed circle.

Q: How do I check my answer?

A: Pick a value from your solution set and substitute it back into the original inequality. If it makes the inequality true, your answer is likely correct.

Exit Presentation

Solving and Graphing Inequalities

1 / 13

Algebra / Inequalities

Solving and Graphing Inequalities

Learn how to solve one-step and two-step inequalities and represent solutions on a number line.

Intermediate25 min

Use arrow keys or spacebar to navigate