Worked Solutions
Level: community college placement test prep, with explanations for someone returning to math after a long break. All 5 problems are covered. Per your request, I go slowly and add extra help wherever negatives or percentages come up.
A Quick Refresher on Negatives (used throughout)
Negative numbers come up in almost every problem below, so here are the rules in one place.
| Situation | Rule | Example |
|---|---|---|
| Adding a negative | Same as subtracting | 5 + (−3) = 2 |
| Subtracting a negative | Same as adding (two minuses make a plus) | 5 − (−3) = 8 |
| Negative + negative | Stay negative, get "more negative" | −10 − 6 = −16 |
| Negative + positive | Subtract the sizes; the bigger one's sign wins | −10 + 6 = −4 |
| Multiplying or dividing, signs match | Answer is positive | (−8) ÷ (−4) = 2 |
| Multiplying or dividing, signs differ | Answer is negative | 3 × (−6) = −18 |
Everyday picture: think of negatives as money you owe. If you owe $10 and pay back $6, you still owe $4, so −10 + 6 = −4.
Problem 1
The problem: Solve for x: 3(x − 4) + 2 = 5x − 6
Key idea: This is a linear equation, meaning x appears only to the first power (no x²). The strategy is to clean up each side, gather all the x's on one side and the plain numbers on the other, then divide. You can spot this type because x appears on both sides and there are parentheses to multiply out.
Step-by-step solution:
3(x − 4) + 2 = 5x − 6
3x − 12 + 2 = 5x − 6 Step 1
3x − 10 = 5x − 6 Step 2
−10 = 2x − 6 Step 3
−4 = 2x Step 4
−2 = x Step 5
- Distribute the 3. "Distribute" means multiply the 3 by each thing inside the parentheses. 3 × x = 3x, and 3 × (−4) = −12. The minus sign stays with the 4 the whole way.
- Combine the plain numbers on the left. −12 + 2: you owe 12, you get 2 back, so you still owe 10. That gives −10.
- Move the x's to one side. Subtract 3x from both sides. I chose to subtract 3x (the smaller x-term) so the x's stay positive, which avoids extra negatives. On the left, 3x − 3x = 0. On the right, 5x − 3x = 2x.
- Move the plain numbers to the other side. The right side has −6. To undo subtracting 6, add 6 to both sides. On the left, −10 + 6 = −4 (owe 10, pay back 6, still owe 4). On the right, −6 + 6 = 0.
- Divide both sides by 2. −4 ÷ 2 = −2. A negative divided by a positive is negative.
Answer: x = −2
Check: Put x = −2 back into the original equation. Both sides should match.
Left side: 3(−2 − 4) + 2 = 3(−6) + 2 = −18 + 2 = −16
Right side: 5(−2) − 6 = −10 − 6 = −16
Both sides equal −16, so x = −2 is correct.
Watch out for:
- Losing a minus sign. It's easy to turn −10 + 6 into +4, or to drop the negative in the last step. Read each sign out loud as "owe" or "have" if it helps.
- Distributing to only the first term. 3(x − 4) is 3x − 12, not 3x − 4.
Your attempt: You got x = 2. That is very close: you have the right size of number, and only the sign is off. That tells me your overall method (distribute, combine, move terms, divide) was probably working. If we plug in x = 2, the left side gives −4 and the right side gives 4. They are mirror images, which is the classic sign of a single slip with a negative.
I can't see your written steps, so I can't point to the exact line. The most likely spot is Step 4: −10 + 6. It's tempting to write 4 here, but owing 10 and paying back 6 leaves you still owing 4, so it's −4. A second possible spot is moving the −6 across and accidentally subtracting it instead of adding it. Next time, try plugging your answer back in. Your "I'm not sure" instinct was a good one, and the check would have caught it.
Problem 2
The problem: Simplify (x² − 9)/(x + 3)
Assumption: I'm reading this as the whole expression x² − 9 divided by the whole expression x + 3.
Key idea: To simplify a fraction with x's in it, first factor the top and bottom. Factoring means rewriting something as pieces multiplied together. Then cancel any piece that appears on both top and bottom. A "something² minus a perfect square" on top is a strong hint. That pattern is called a difference of squares.
Step-by-step solution:
(x² − 9)/(x + 3)
= (x − 3)(x + 3)/(x + 3) Step 1–2
= x − 3 Step 3
- Recognize the pattern. x² is x times x, and 9 is 3 times 3. So x² − 9 is "a square minus a square."
- Factor it with the difference-of-squares rule: a² − b² = (a − b)(a + b). With a = x and b = 3, we get x² − 9 = (x − 3)(x + 3). You can confirm this by multiplying back out: x·x + 3x − 3x − 9 = x² − 9. The middle terms cancel each other.
- Cancel the matching factor. (x + 3) is now multiplied on top and sits on the bottom, so they divide to 1. That's like (5 × 7)/7 = 5.
- Note one restriction. The original fraction can't have a zero on the bottom, so x can't be −3. The simplified version keeps that rule: x − 3, for x ≠ −3. Many placement tests just want x − 3, but it's good to know why the restriction exists.
Answer: x − 3 (for x ≠ −3)
Check: Pick any allowed number, say x = 5, and compare the original with the simplified version.
Original: (5² − 9)/(5 + 3) = (25 − 9)/8 = 16/8 = 2
Simplified: 5 − 3 = 2
They match.
Watch out for:
- Cancelling pieces that are added or subtracted, rather than multiplied. You can only cancel factors (things multiplied), never terms (things added or subtracted).
- Forgetting to factor first. If you don't factor, nothing looks cancellable, or you cancel the wrong things.
Your attempt: Your sense that x² − 3 "seems wrong" was exactly right. Trust that instinct. The problem was cancelling the x's. In (x² − 9)/(x + 3), the x on the bottom is added to 3, not multiplied, so it can't be cancelled on its own.
Here is an everyday version of why.