Algebra 1
Foundations & the Language of Algebra
Fresh, parallel-form problems with full worked solutions — more reps for the skills in this unit, kept separate from the textbook's own problems.
Lesson 1.1: Variables and the meaning of equations
Solve by undoing (show the undo step): x + 5 = 12.
Worked solutionTry it first, then open.
The 5 is added on, so undo it by subtracting 5 from both sides to keep the balance: x + 5 - 5 = 12 - 5, which gives x = 7. Check: 7 + 5 = 12.
Answer: 7
Solve by undoing (show the undo step): 3x = 21.
Worked solutionTry it first, then open.
The x is multiplied by 3, so undo by dividing both sides by 3: (3x)/3 = 21/3, which gives x = 7. Check: 3·7 = 21.
Answer: 7
Solve by undoing (show the undo step): x/4 = 6.
Worked solutionTry it first, then open.
The x is divided by 4, so undo by multiplying both sides by 4: (x/4)·4 = 6·4, which gives x = 24. Check: 24/4 = 6. (Multiplying undoes dividing — the opposite move.)
Answer: 24
Fill the blank so the sentence stays true: 9 + 6 = □ + 4. (Read = as "the same as" — both sides must balance.)
Worked solutionTry it first, then open.
Remember = means "the same as," not "compute here." The left side is 9 + 6 = 15, so the right side must also equal 15: □ + 4 = 15, which means □ = 11. Check: 9 + 6 = 11 + 4 → 15 = 15. (Filling 15 is the "compute here" misread — it ignores the + 4 still sitting on the right pan.)
Answer: 11
Lesson 1.2: The number system & the number line
Classify -9. Label all types that apply (natural / whole / integer / rational / irrational / real), and note non-integer where it matters.
Worked solutionTry it first, then open.
-9 is a whole amount with no fractional part and it is negative, so it is an integer. Every integer is a ratio of integers (-9 = -9/1), so it is also rational. And like every number we meet here, it is real. It is not natural or whole, since those start at 0 and go up. So: integer, rational, real.
Answer: integer, rational, real
Classify 2.5. Label all types that apply, and note non-integer where it matters.
Worked solutionTry it first, then open.
2.5 is not a whole amount, so it is not an integer. But it can be written as a ratio of integers: 2.5 = 5/2. That makes it rational (a non-integer rational), and therefore real. So: rational, non-integer, real.
Answer: rational, non-integer, real
Classify 12/3. Don't be fooled by the fraction bar — evaluate first, then classify.
Worked solutionTry it first, then open.
Evaluate before you classify: 12/3 = 4. Its value is 4, a whole amount with no fractional part, so it is an integer (and rational, since 4 = 4/1, and real). How a number is written — even as a fraction — does not fix its type; its value does. So: =4 → integer, rational, real.
Answer: =4 → integer, rational, real
Classify √7. Is it rational or irrational? Then contrast: what about √25? (Light stretch — spelling vs. value.)
Worked solutionTry it first, then open.
√7 ≈ 2.6457513… — its decimal never ends and never repeats, so it cannot be written as a ratio of two integers. That makes it irrational (and, like every point on the line, real). Contrast √25: it looks like a root, but √25 = 5, an integer — value, not spelling, decides. So √7 is irrational, real; √25 is an integer.
Answer: irrational, real
Lesson 1.3: Order of operations
Evaluate using order of operations: 6 + 4 × 2.
Worked solutionTry it first, then open.
Multiply/divide is a higher tier than add/subtract, so multiply first: 4 × 2 = 8. Then add: 6 + 8 = 14. (Not 20 — you don't add before multiplying.)
Answer: 14
Evaluate using order of operations: 24 ÷ 4 × 3.
Worked solutionTry it first, then open.
Multiply and divide are equal-rank partners, so work left to right: 24 ÷ 4 = 6, then 6 × 3 = 18. (Not 2 — there is no "multiply before divide"; left to right within the tier.)
Answer: 18
Evaluate using order of operations: 5 + 2 × 3².
Worked solutionTry it first, then open.
Exponents are a higher tier than multiply, so square first: 3² = 9. Then multiply: 2 × 9 = 18. Then add: 5 + 18 = 23. (Squaring 2×3 first to get 36 is the trap — the exponent acts before the multiply.)
Answer: 23
Evaluate using order of operations: (9 − 3)² ÷ 4 + 2 × 5. (Light stretch — all four tiers.)
Worked solutionTry it first, then open.
Grouping first: (9 − 3) = 6. Then the exponent: 6² = 36. Now the multiply/divide tier, left to right: 36 ÷ 4 = 9, and 2 × 5 = 10. Finally the add/subtract tier: 9 + 10 = 19.
Answer: 19
Lesson 1.4: Factors and exponents
List all the factors of 30.
Worked solutionTry it first, then open.
Walk up from 1 and keep the ones that divide 30 cleanly: 1, 2, 3, 5, 6, 10, 15, 30. They pair up to 30 — 1·30, 2·15, 3·10, 5·6 — which is how you know the list is complete.
Answer: 1, 2, 3, 5, 6, 10, 15, 30
Is 13 prime or composite? Say how you know.
Worked solutionTry it first, then open.
Look for a factor other than 1 and 13: 2, 3, 4, 5, and 6 each leave a remainder, and past 6 the partners only repeat. So 13's only factors are 1 and itself — exactly two — which makes it prime.
Answer: prime
Evaluate 3³.
Worked solutionTry it first, then open.
The base is 3 and the exponent is 3, so 3 is multiplied three times: 3³ = 3 · 3 · 3 = 27. (The exponent counts the factors of 3; it does not mean 3 × 3.)
Answer: 27
Write the prime factorization of 24, using an exponent where a prime repeats. (Light stretch.)
Worked solutionTry it first, then open.
Break 24 down and keep splitting until only primes remain: 24 = 4 · 6 = (2 · 2)(2 · 3) = 2 · 2 · 2 · 3. The 2 repeats three times, so stack it under an exponent: 24 = 2³ · 3.
Answer: 2³ · 3
Lesson 1.5: Negative numbers
Evaluate: -6 + (-8).
Worked solutionTry it first, then open.
Adding a negative moves further left on the number line: start at -6 and move 8 more to the left, landing at -14. (Two debts pile up: owing 6 and then owing 8 more means owing 14.) So -6 + (-8) = -14.
Answer: -14
Evaluate: 7 - (-5).
Worked solutionTry it first, then open.
Subtracting a negative is the same as adding its opposite: 7 - (-5) = 7 + 5 = 12. (A debt forgiven leaves you richer — two about-faces.) So 7 - (-5) = 12.
Answer: 12
Evaluate: (-4)(-3)(2).
Worked solutionTry it first, then open.
Count the negative factors: there are two of them, an even number, so the product is positive. Multiply the sizes: 4 × 3 × 2 = 24. So (-4)(-3)(2) = 24.
Answer: 24
Evaluate: -3² + (-2)(-5). (Light stretch — mind the exponent's reach and the sign rules.)
Worked solutionTry it first, then open.
The exponent touches only the 3, and the leading minus is applied after squaring: -3² = -(3²) = -(9) = -9. Next, (-2)(-5) = 10, since same signs give a positive. Then add: -9 + 10 = 1. (If you got 9 + 10 = 19, you squared the negative — but without parentheses the minus comes after the square.)
Answer: 1
Lesson 1.6: Expressions vs. equations; evaluating expressions
Is 4x - 7 an expression or an equation? How do you know, and what would you do with it?
Worked solutionTry it first, then open.
There is no equals sign, so it is an expression — a phrase that names a value once you know x. You don't "solve" it; you evaluate it by substituting a number for x. (An equation, by contrast, has an = and makes a claim you solve, like 4x - 7 = 5.) So: expression.
Answer: expression
Evaluate the expression 2x + 5 at x = 3.
Worked solutionTry it first, then open.
Substitute 3 for x — the reserved seat is filled: 2(3) + 5. Order of operations: multiply first, 2(3) = 6, then add, 6 + 5 = 11. So 2x + 5 at x = 3 is 11.
Answer: 11
Evaluate the expression 6 - 3x at x = -2. (Keep parentheses around the substituted value.)
Worked solutionTry it first, then open.
Substitute -2 for x, keeping it in parentheses: 6 - 3(-2). Multiply first: 3(-2) = -6, so the expression is 6 - (-6). Subtracting a negative adds: 6 + 6 = 12. So 6 - 3x at x = -2 is 12.
Answer: 12
Evaluate the expression x² - 2x at x = -3. (Light stretch — exponent on a negative, then a signed product.)
Worked solutionTry it first, then open.
Substitute -3 for x, in parentheses: (-3)² - 2(-3). The whole -3 is squared because of the parentheses: (-3)² = 9. Next, 2(-3) = -6, so we have 9 - (-6). Subtracting a negative adds: 9 + 6 = 15. So x² - 2x at x = -3 is 15.
Answer: 15
Mixed review
Problems that mix skills from across the unit — good for spacing earlier work back in.
Solve by undoing (show the undo step): 4x = 28.
Worked solutionTry it first, then open.
The x is multiplied by 4, so undo by dividing both sides by 4: (4x)/4 = 28/4, giving x = 7. Check: 4·7 = 28. (Lesson 1.1: undo the operation attached to the variable, same move to both sides.)
Answer: 7
Evaluate: -5 - (-9).
Worked solutionTry it first, then open.
Subtracting a negative is adding its opposite: -5 - (-9) = -5 + 9. Start at -5 and move 9 to the right, landing at 4. So -5 - (-9) = 4. (Lesson 1.5: separate the operation sign from the number's sign.)
Answer: 4
Evaluate using order of operations: 3 + 2 × (7 − 4).
Worked solutionTry it first, then open.
Grouping first: (7 − 4) = 3. Then multiply: 2 × 3 = 6. Then add: 3 + 6 = 9. So the value is 9. (Lesson 1.3: parentheses, then the multiply/divide tier, then add/subtract.)
Answer: 9
Evaluate the expression 10 - 2x at x = -3. (Keep parentheses around the substituted value.)
Worked solutionTry it first, then open.
Substitute -3 for x, in parentheses: 10 - 2(-3). Multiply first: 2(-3) = -6, so we have 10 - (-6). Subtracting a negative adds: 10 + 6 = 16. So 10 - 2x at x = -3 is 16. (Lessons 1.6 + 1.5 + 1.3 together: substitute, order of operations, signed arithmetic.)
Answer: 16
Evaluate: (-2)³ + 4 × 5. (Light stretch — interleaves exponents on a negative, the sign rules, and order of operations.)
Worked solutionTry it first, then open.
The parentheses mean the whole -2 is cubed: (-2)³ = (-2)(-2)(-2). Three negative factors is an odd count, so the result is negative: (-2)³ = -8. Next, multiply: 4 × 5 = 20. Then add: -8 + 20 = 12. So (-2)³ + 4 × 5 = 12. (Lessons 1.3 + 1.5: exponent tier first, then multiply, then add; count the negatives.)
Answer: 12