Addition and Multiplication

Solve: $3x + 7 > 22$

Identify the operations on x: $x$ is multiplied by 3, then 7 is added = Operations: ×3, then +7

Undo addition first: Subtract 7 from both sides: $3x + 7 - 7 > 22 - 7$ = $3x > 15$

Undo multiplication: Divide both sides by 3: $\frac{3x}{3} > \frac{15}{3}$ = $x > 5$

Check with a value: Try $x = 6$: $3(6) + 7 = 25$, and $25 > 22$ ✓ = Solution verified

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

Subtraction and Division

Solve: $\frac{x}{4} - 3 \leq 2$

Identify the operations: $x$ is divided by 4, then 3 is subtracted = Operations: ÷4, then −3

Undo subtraction first: Add 3 to both sides: $\frac{x}{4} - 3 + 3 \leq 2 + 3$ = $\frac{x}{4} \leq 5$

Undo division: Multiply both sides by 4: $\frac{x}{4} \times 4 \leq 5 \times 4$ = $x \leq 20$

Check boundary value: Try $x = 20$: $\frac{20}{4} - 3 = 2$, and $2 \leq 2$ ✓ = Boundary included

Answer: $x \leq 20$. The solution includes 20 and all smaller numbers.

Negative Coefficient (FLIP Required!)

Solve: $-2x + 8 < 14$

Identify the operations: $x$ is multiplied by $-2$, then 8 is added = Operations: ×(−2), then +8

Undo addition first: Subtract 8 from both sides: $-2x + 8 - 8 < 14 - 8$ = $-2x < 6$

Divide by NEGATIVE: Divide both sides by $-2$: $\frac{-2x}{-2}$ ? $\frac{6}{-2}$ = Need to FLIP!

Apply the flip rule: $<$ becomes $>$ when dividing by negative = $x > -3$

Verify the flip: Try $x = 0$: $-2(0) + 8 = 8$, and $8 < 14$ ✓ = Flip confirmed!

Answer: $x > -3$. Because we divided by a negative number, we flipped the sign!

Constant Term First

Solve: $5 - 4x \geq 17$

Rewrite in standard form: Notice: constant (5) comes first, then $-4x$ = $5 - 4x \geq 17$

Isolate the variable term: Subtract 5 from both sides: $-4x \geq 12$ = $-4x \geq 12$

Divide by NEGATIVE coefficient: Divide by $-4$ and FLIP: $x \leq \frac{12}{-4}$ = $x \leq -3$

Check the solution: Try $x = -4$: $5 - 4(-4) = 5 + 16 = 21$, and $21 \geq 17$ ✓ = Verified!

Answer: $x \leq -3$. We flipped $\geq$ to $\leq$ because we divided by $-4$.

Fraction with Two Steps

Solve: $\frac{2x - 6}{3} > 4$

Eliminate the denominator: Multiply both sides by 3: $2x - 6 > 12$ = No flip (3 is positive)

Undo subtraction: Add 6 to both sides: $2x > 18$ = $2x > 18$

Undo multiplication: Divide by 2: $x > 9$ = $x > 9$

Verify: Try $x = 10$: $\frac{2(10) - 6}{3} = \frac{14}{3} \approx 4.67$, and $4.67 > 4$ ✓ = Verified!

Answer: $x > 9$. When clearing fractions, multiply all terms by the denominator.

Variable on the Right

Solve: $10 < 3 - 7x$

Strategy: We can solve as-is, or rewrite with variable on left = Solving as written

Isolate the term with x: Subtract 3 from both sides: $7 < -7x$ = $7 < -7x$

Divide by NEGATIVE: Divide by $-7$ and FLIP: $\frac{7}{-7} > x$ = $-1 > x$

Rewrite with x on left: $-1 > x$ is the same as $x < -1$ = $x < -1$

Answer: $x < -1$. You can solve with the variable on either side, just be careful with the flip!

Mistake: Undoing operations in the wrong order

Why: Students try to divide first instead of handling addition/subtraction first.

Correct: Always undo addition/subtraction BEFORE multiplication/division. Think PEMDAS in reverse!

Mistake: Forgetting to flip when dividing by a negative

Why: After multiple steps, students lose track of whether they're dividing by positive or negative.

Correct: Circle the coefficient! If it's negative and you're dividing, put a big FLIP reminder.

Mistake: Only applying operations to one side

Why: With two steps, students sometimes forget to apply the second operation to both sides.

Correct: Whatever you do to the left side, you MUST do to the right side. Check both sides each step.

Mistake: Distributing incorrectly with negative signs

Why: In problems like $5 - 4x \geq 17$, students mishandle the negative coefficient.

Correct: Keep track of signs carefully. $5 - 4x$ means $+5$ and $-4x$. The coefficient of $x$ is $-4$.

Phone Plan Budget

Mobile plans often have a fixed monthly fee plus charges for extra data or minutes.

A plan costs 25 euros per month plus 3 euros per GB of extra data. If you can spend at most 40 euros: $25 + 3g \leq 40$, so $g \leq 5$ GB.

Saving for a Purchase

When you want to buy something but also need to keep a minimum balance.

You have 180 euros in savings. After weekly expenses of 12 euros, you want at least 60 euros left. How many weeks until you must stop? $180 - 12w \geq 60$.

Temperature Requirements

Industrial processes often require temperatures within specific ranges.

A chemical must be heated from 20°C, rising 5°C per minute, but must stay below 80°C. How long can you heat it? $20 + 5t < 80$, so $t < 12$ minutes.

Q: What's the difference between one-step and two-step inequalities?

A: One-step inequalities need only one operation (like $x + 5 > 8$). Two-step inequalities need two operations in sequence (like $2x + 5 > 11$).

Q: Do I flip the sign at every step?

A: No! Only flip when you MULTIPLY or DIVIDE by a negative number. Adding or subtracting negatives does NOT require a flip.

Q: What if the variable has a negative coefficient like $-3x$?

A: When you isolate $x$ by dividing both sides by $-3$, you MUST flip the inequality sign because $-3$ is negative.

Q: Can I add or subtract first if multiplication seems easier?

A: Technically yes, but it usually makes things harder. Following the standard order (undo +/− first, then ×/÷) is more reliable.