College Algebra Quiz: Linear Inequalities And Interval Notation

What this quiz covers

This quiz focuses on Linear Inequalities And Interval Notation, giving you a quick way to practice the rules, question types, and explanations that matter most for College Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $( -\infty,45]$ (correct answer)
2. $[45,\infty)$
3. $( -\infty,45)$
4. $[ -\infty,45]$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 3x + 45 ≤ 180 requires isolating the variable and determining the solution set, represented as (-∞, 45]. The correct choice, A, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it misapplies the inequality direction, often confusing students about reversing the symbol when necessary. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 2

1. $(-\infty,3]$
2. $(-\infty,3)$
3. $[3,\infty)$ (correct answer)
4. $(3,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 10 ≤ 2t + 4 requires isolating the variable and determining the solution set, represented as [3, ∞). The correct choice, C, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it uses an open interval, often confusing students about inclusive inequalities. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 3

1. $(-\infty,100)$ (correct answer)
2. $(-\infty,100]$
3. $(100,\infty)$
4. $[100,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 0.5x + 30 < 80 requires isolating the variable and determining the solution set, represented as (-∞, 100). The correct choice, A, accurately reflects this solution by properly applying interval notation standards. A common distractor, D, fails because it includes the endpoint incorrectly, often confusing students about strict inequalities. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 4

1. $[-3,\infty)$
2. $(-\infty,-3)$
3. $( -\infty,-3]$ (correct answer)
4. $(-3,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality -6x ≥ 18 requires isolating the variable and determining the solution set, represented as (-∞, -3]. The correct choice, C, accurately reflects this solution by properly applying interval notation standards. A common distractor, A, fails because it omits the endpoint, often confusing students about inequality reversal with negatives. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 5

1. $(-\infty,2)$
2. $[2,\infty)$
3. $(-\infty,2]$ (correct answer)
4. $(2,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality - (s/2) + 1 ≥ 0 requires isolating the variable and determining the solution set, represented as (-∞, 2]. The correct choice, C, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it uses the wrong direction, often confusing students about negative coefficients. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 6

1. $(-\infty,8)$
2. $(8,\infty)$ (correct answer)
3. $(-\infty,8]$
4. $[8,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 5 - t/2 < 1 requires isolating the variable and determining the solution set, represented as (8, ∞). The correct choice, B, accurately reflects this solution by properly applying interval notation standards. A common distractor, A, fails because it uses the wrong inequality direction, often confusing students about multiplying by negatives. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 7

1. $(-\infty,3]$
2. $(-\infty,3)$
3. $[3,\infty)$ (correct answer)
4. $(3,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 9 ≤ 3s requires isolating the variable and determining the solution set, represented as [3, ∞). The correct choice, C, accurately reflects this solution by properly applying interval notation standards. A common distractor, A, fails because it uses a closed interval in the wrong direction, often confusing students about dividing positives. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 8

1. $(-\infty,-6)$
2. $(-\infty,-6]$
3. $(-6,\infty)$ (correct answer)
4. $[-6,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality (x+9)/3 > 1 requires isolating the variable and determining the solution set, represented as (-6, ∞). The correct choice, C, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it includes equality, often confusing students about strict inequalities. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 9

1. $(-\infty,6)$
2. $(6,\infty)$ (correct answer)
3. $(-\infty,6]$
4. $[6,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality - (x/3) + 4 < 2 requires isolating the variable and determining the solution set, represented as (6, ∞). The correct choice, B, accurately reflects this solution by properly applying interval notation standards. A common distractor, A, fails because it reverses the direction, often confusing students about negative coefficients. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 10

1. $(-\infty,3)$ (correct answer)
2. $(-\infty,3]$
3. $(3,\infty)$
4. $[3,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality -3x + 9 > 0 requires isolating the variable and determining the solution set, represented as (-∞, 3). The correct choice, A, accurately reflects this solution by properly applying interval notation standards. A common distractor, D, fails because it misapplies the inequality flip, often confusing students about direction when multiplying by negatives. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 11

1. $(-\infty,-14)$
2. $(-14,\infty)$ (correct answer)
3. $[-14,\infty)$
4. $(-\infty,-14]$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality -2(x+5) < 18 requires isolating the variable and determining the solution set, represented as (-14, ∞). The correct choice, B, accurately reflects this solution by properly applying interval notation standards. A common distractor, D, fails because it includes a bracket incorrectly, often confusing students about strict inequalities after flipping. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 12

1. $(-\infty,5)$ (correct answer)
2. $(-\infty,5]$
3. $[5,\infty)$
4. $(5,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 15 > 5x - 10 requires isolating the variable and determining the solution set, represented as (-∞, 5). The correct choice, A, accurately reflects this solution by properly applying interval notation standards. A common distractor, C, fails because it includes equality incorrectly, often confusing students about strict inequalities. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 13

1. $(-\infty,12]$
2. $[12,\infty)$ (correct answer)
3. $(-\infty,12)$
4. $(12,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 10 - x/2 ≤ 4 requires isolating the variable and determining the solution set, represented as [12, ∞). The correct choice, B, accurately reflects this solution by properly applying interval notation standards. A common distractor, D, fails because it excludes the endpoint, often confusing students when flipping with fractions. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 14

1. $(-\infty,-4]$
2. $[-4,\infty)$ (correct answer)
3. $(-4,\infty)$
4. $(-\infty,-4)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality -4x + 12 ≤ 28 requires isolating the variable and determining the solution set, represented as [-4, ∞). The correct choice, B, accurately reflects this solution by properly applying interval notation standards. A common distractor, A, fails because it reverses the inequality direction, often confusing students when dividing by negative numbers. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 15

1. $(-\infty,4)$
2. $[4,\infty)$
3. $(-\infty,4]$ (correct answer)
4. $(4,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 9 - 2x ≥ 1 requires isolating the variable and determining the solution set, represented as (-∞, 4]. The correct choice, C, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it reverses the direction, often confusing students when flipping inequalities with negatives. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 16

1. $S = [-3, 2]$ and includes all endpoints
2. $S = (-3, 2]$ and excludes $x = -3$ (correct answer)
3. $S = [-3, 2)$ and excludes $x = 2$
4. $S = (-3, 2)$ and excludes both endpoints

Explanation: For $\frac{x-2}{x+3} \leq 0$, we need the fraction to be negative or zero. The critical points are $x = 2$ (numerator zero) and $x = -3$ (denominator zero). We must exclude $x = -3$ since it makes the denominator zero. Testing intervals: For $x < -3$: both numerator and denominator are negative, so the fraction is positive. For $-3 < x < 2$: numerator is negative, denominator is positive, so the fraction is negative. For $x > 2$: both are positive, so the fraction is positive. At $x = 2$: the fraction equals zero. Therefore, $S = (-3, 2]$. Choice A incorrectly includes $x = -3$. Choice C incorrectly excludes $x = 2$. Choice D excludes both endpoints incorrectly.

Question 17

1. $[50, 160]$
2. $(50, 160)$
3. $[50, 160)$ (correct answer)
4. $(50, 160]$

Explanation: The cost function is $C(x) = 12x + 480$. We need $C(x) < 2400$ and $x \geq 50$. Solving $12x + 480 < 2400$: $12x < 1920$, so $x < 160$. Combined with $x \geq 50$, we get $50 \leq x < 160$, which is $[50, 160)$. Choice A includes 160 (inequality should be strict). Choice B excludes 50 (should be included). Choice D includes 160 and excludes 50 (both wrong).

Question 18

1. $(-\infty,2)$
2. $[2,\infty)$ (correct answer)
3. $(-\infty,2]$
4. $(2,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 12 ≤ 3x + 6 requires isolating the variable and determining the solution set, represented as [2, ∞). The correct choice, B, accurately reflects this solution by properly applying interval notation standards. A common distractor, A, fails because it reverses the direction, often confusing students about rearranging terms. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 19

1. $(-\infty,5]$
2. $(5,\infty)$
3. $(-\infty,5)$ (correct answer)
4. $[5,\infty)$

Explanation: This question tests college algebra skills in solving linear inequalities and expressing solutions in interval notation. Linear inequalities involve expressions where one side is greater or lesser than the other; interval notation succinctly represents these solutions on the number line. In this specific scenario, solving the inequality 7x - 14 < 21 requires isolating the variable and determining the solution set, represented as (-∞, 5). The correct choice, C, accurately reflects this solution by properly applying interval notation standards. A common distractor, D, fails because it reverses the inequality, often confusing students about the direction of the solution. To assist students, emphasize the importance of understanding open vs. closed intervals and the correct application of inequality symbols. Practice solving inequalities and converting results into interval notation to build fluency.

Question 20

1. Confused the direction of the inequality symbol from the beginning
2. Made an arithmetic error when combining like terms
3. Incorrectly distributed the negative sign across terms
4. Failed to flip the inequality sign when dividing by a negative coefficient (correct answer)

Explanation: When solving linear inequalities, you follow the same steps as solving equations, with one crucial exception: you must flip the inequality sign whenever you multiply or divide both sides by a negative number. Let's solve this inequality correctly: $-4x + 12 < 2x - 6$. First, subtract $2x$ from both sides: $-6x + 12 < -6$. Then subtract 12 from both sides: $-6x < -18$. Now comes the critical step: divide both sides by $-6$. Since we're dividing by a negative number, we must flip the inequality sign: $x > 3$. The student got $x < 3$, which suggests they correctly performed all the algebraic steps but forgot to flip the inequality sign when dividing by $-6$. This confirms answer choice D. Let's examine why the other options don't fit. Choice A suggests the student confused the inequality direction from the start, but their intermediate steps would have been completely different. Choice B (arithmetic errors with like terms) and choice C (incorrect distribution of negative signs) would likely produce different final answers, not just the wrong inequality direction with the correct boundary value of 3. The key takeaway: always remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. This is because negative multiplication flips the relative positions of numbers on the number line. Make this your automatic check whenever you see a negative coefficient in front of your variable.