Addition Inequality

Solve: $x + 7 > 12$

Identify the operation: 7 is being added to $x$ = Operation: addition

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

Check with a value: Try $x = 6$: $6 + 7 = 13$, and $13 > 12$ ✓ = Solution verified

Answer: $x > 5$. Any number greater than 5 is a solution.

Subtraction Inequality

Solve: $x - 4 \leq 9$

Identify the operation: 4 is being subtracted from $x$ = Operation: subtraction

Use inverse operation: Add 4 to both sides: $x - 4 + 4 \leq 9 + 4$ = $x \leq 13$

Check boundary value: Try $x = 13$: $13 - 4 = 9$, and $9 \leq 9$ ✓ = Boundary included

Answer: $x \leq 13$. Any number 13 or less is a solution.

Multiplication Inequality (Positive)

Solve: $5x \geq 35$

Identify the operation: $x$ is being multiplied by 5 = Operation: multiplication by positive

Use inverse operation: Divide both sides by 5: $\frac{5x}{5} \geq \frac{35}{5}$ = $x \geq 7$

Verify (no sign flip): Dividing by positive 5, so inequality sign stays the same = No flip needed

Answer: $x \geq 7$. The solution is 7 or any number greater.

Division Inequality

Solve: $\frac{x}{3} < 4$

Identify the operation: $x$ is being divided by 3 = Operation: division by positive

Use inverse operation: Multiply both sides by 3: $\frac{x}{3} \times 3 < 4 \times 3$ = $x < 12$

Verify (no sign flip): Multiplying by positive 3, so inequality sign stays the same = No flip needed

Answer: $x < 12$. Any number less than 12 is a solution.

Multiplication by Negative (FLIP!)

Solve: $-3x > 15$

Identify the operation: $x$ is being multiplied by $-3$ = Operation: multiplication by NEGATIVE

Divide by negative: Divide both sides by $-3$: $\frac{-3x}{-3}$ ? $\frac{15}{-3}$ = Need to flip!

FLIP the inequality: Dividing by negative: $>$ becomes $<$ = $x < -5$

Verify the flip: Try $x = -6$: $-3(-6) = 18$, and $18 > 15$ ✓ = Flip confirmed!

Answer: $x < -5$. Because we divided by a negative number, we flipped the sign!

Division by Negative (FLIP!)

Solve: $\frac{x}{-2} \geq 6$

Identify the operation: $x$ is being divided by $-2$ = Operation: division by NEGATIVE

Multiply by negative: Multiply both sides by $-2$: $\frac{x}{-2} \times (-2)$ ? $6 \times (-2)$ = Need to flip!

FLIP the inequality: Multiplying by negative: $\geq$ becomes $\leq$ = $x \leq -12$

Verify the flip: Try $x = -14$: $\frac{-14}{-2} = 7$, and $7 \geq 6$ ✓ = Flip confirmed!

Answer: $x \leq -12$. Multiplying by $-2$ means we flip $\geq$ to $\leq$.

Why Does the Sign Flip?

Demonstrate why $-x > 3$ becomes $x < -3$

Start with known fact: We know $5 > 2$ is TRUE = 5 is greater than 2

Multiply both by $-1$: $5 \times (-1)$ ? $2 \times (-1)$ = $-5$ ? $-2$

Compare the results: Is $-5 > -2$? NO! $-5 < -2$ = The order reversed!

Apply to original: $-x > 3 \Rightarrow$ multiply by $-1$ and flip: $x < -3$ = $x < -3$

Answer: Multiplying by a negative reverses the order of numbers on the number line, so we must flip the inequality sign.

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

Why: This is the most common error! Students apply inverse operations correctly but forget this special rule.

Correct: ALWAYS check: Am I multiplying or dividing by a NEGATIVE? If yes, FLIP the inequality sign!

Mistake: Flipping the sign for any operation with negatives

Why: Students over-apply the rule, flipping when adding or subtracting negative numbers too.

Correct: Only flip when MULTIPLYING or DIVIDING by a negative. Adding $x + (-5) > 3$ does NOT require a flip!

Mistake: Doing different operations to each side

Why: Unlike equations, the visual difference in inequalities can confuse students.

Correct: Just like equations: whatever you do to the left side, you MUST do to the right side too.

Mistake: Writing the answer in the wrong direction

Why: After solving, students write $5 > x$ instead of $x < 5$.

Correct: Convention is to write the variable on the left: $x < 5$ is clearer than $5 > x$.

Budgeting and Saving

When you need to save a minimum amount, inequalities help determine how much you can spend.

You have 75 euros and want to keep at least 30 euros for emergencies. How much can you spend? $75 - x \geq 30$, so $x \leq 45$.

Temperature Limits

Science experiments often require temperatures within certain ranges.

A chemical reaction requires the temperature to stay below 100°C. If the room is 25°C and heat is added at 15° per minute: $25 + 15t < 100$.

Sports and Games

Determining what scores are needed to win or qualify.

Your team has 48 points. Each basket is worth 2 points. How many more baskets to reach 60? $48 + 2b \geq 60$, so $b \geq 6$.

Q: Why do we flip the inequality when multiplying by a negative?

A: When you multiply numbers by $-1$, their order reverses. For example, $3 > 2$ but $-3 < -2$. The flip keeps the inequality TRUE after the operation.

Q: Do I flip for adding or subtracting negative numbers?

A: No! Only flip when MULTIPLYING or DIVIDING by a negative. Adding $-5$ to both sides does not flip anything.

Q: How do I check my answer?

A: Pick any number from your solution set and plug it into the original inequality. If it works, you're correct! Also try a number NOT in the solution set to confirm it fails.

Q: What if the variable is on the right side?

A: You can solve it normally, then rewrite with the variable on the left. Remember: $5 > x$ is the same as $x < 5$.