Algebra 1 Practice Test: Solving Linear Equations and Inequalities
- Solving Linear Equations and Inequalities
- 15 questions
- 20 minutes
These 15 questions cover multi-step linear equations, equations with no or infinitely many solutions, literal equations, word problems, compound inequalities and absolute value. When you multiply or divide an inequality by a negative number, reverse the inequality sign.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What is the solution of 3(x − 4) + 2 = 2x + 5?
Correct answer: D
Distribute first: 3x − 12 + 2 = 2x + 5, so x − 10 = 5 and x = 15. Writing 3(x − 4) as 3x − 4 leads to 7.
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Question 2
Solve 2x3 − 1 = 7.
Correct answer: A
Add 1 to get 2x3 = 8, multiply by 3 to get 2x = 24, then divide by 2: x = 12. Stopping at 2x = 24 gives the trap answer 24.
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Question 3
Which inequality is equivalent to −4x + 7 > 19?
Correct answer: B
Subtract 7: −4x > 12. Dividing by −4 reverses the sign, so x < −3. Forgetting to reverse the sign gives x > −3.
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Question 4
Which equation has no solution?
Correct answer: C
2(x + 3) = 2x + 6, so 2x + 6 = 2x + 5 would require 6 = 5, which is never true. 2(x + 3) = 2x + 6 is true for every x, and the other two each have exactly one solution.
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Question 5
The perimeter of a rectangle is P = 2l + 2w. Which equation is solved for w?
Correct answer: A
Subtract 2l from both sides to get P − 2l = 2w, then divide both sides by 2. Dividing only P by 2 and leaving 2l unchanged is a common error.
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Question 6
A gym charges a one-time sign-up fee of $35 plus $22 per month. Leo has paid $255 in total. For how many months has he been a member?
Correct answer: A
Solve 35 + 22m = 255: 22m = 220, so m = 10. Dividing 255 by 22 without removing the fee gives about 11.6, which is not a whole number of months.
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Question 7
Which inequality is equivalent to −3 ≤ 2x + 5 < 11?
Correct answer: B
Subtract 5 from all three parts: −8 ≤ 2x < 6. Divide all parts by 2: −4 ≤ x < 3. The inequality symbols stay as they are because 2 is positive.
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Question 8
What is the solution set of |2x − 3| = 9?
Correct answer: D
Set 2x − 3 = 9 (so x = 6) and 2x − 3 = −9 (so x = −3). Using only the positive case misses the solution −3.
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Question 9
Which inequality is equivalent to |x − 4| < 5?
Correct answer: A
A distance less than 5 means −5 < x − 4 < 5; adding 4 gives −1 < x < 9. The "or" form describes |x − 4| > 5.
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Question 10
The graph shows the solution set of an inequality.
Which inequality is graphed?
Correct answer: C
The closed (filled) circle means −1 is included, and the shading goes to the right, toward larger numbers. An open circle would be needed for x > −1.
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Question 11
The sum of three consecutive odd integers is 87. What is the largest of the three integers?
Correct answer: C
Let the integers be n, n + 2 and n + 4. Then 3n + 6 = 87, so n = 27, and the integers are 27, 29 and 31. 27 is the smallest, not the largest.
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Question 12
Solve 0.4x + 1.5 = 0.9x − 2.
Correct answer: D
Subtract 0.4x and add 2: 3.5 = 0.5x, so x = 7. Check: 0.4(7) + 1.5 = 4.3 and 0.9(7) − 2 = 4.3.
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Question 13
Maya has $60. She buys a book for $18 and wants to spend the rest on notebooks that cost $3.50 each. What is the greatest number of notebooks she can buy?
Correct answer: B
Solve 18 + 3.5n ≤ 60: 3.5n ≤ 42, so n ≤ 12. The answer 17 ignores the cost of the book (60 / 3.5 ≈ 17.1).
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Question 14
Solve x + 24 = x − 13.
Correct answer: B
Cross-multiply: 3(x + 2) = 4(x − 1), so 3x + 6 = 4x − 4 and x = 10. Check: both sides equal 3.
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Question 15
For what value of k does the equation 3(2x − k) = 6x + 12 have infinitely many solutions?
Correct answer: D
The left side is 6x − 3k. The equation is true for every x when −3k = 12, that is k = −4. With k = 4 the constants would be −12 and 12, so there would be no solution.
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