College Algebra Quiz: Linear Inequalities And Interval Notation

20 questions · exam conditions

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Linear Inequalities And Interval NotationQuestion 1 of 20

Traffic flow: Determine the solution set for $10\le2t+4$ and write it in interval notation.

(-\infty,3]

(-\infty,3)

[3,\infty)

(3,\infty)

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College Algebra Quiz

College Algebra Quiz: Linear Inequalities And Interval Notation

Practice Linear Inequalities And Interval Notation in College Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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.

All questions

Question 1

Traffic flow: Determine the solution set for $10\le2t+4$ and write it in interval notation.

$(-\infty,3]$
$(-\infty,3)$
$[3,\infty)$ (correct answer)
$(3,\infty)$

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 2

Budgeting: Solve $-3x+9>0$ for a discount rate; write solution in interval notation.

$(-\infty,3)$ (correct answer)
$(-\infty,3]$
$(3,\infty)$
$[3,\infty)$

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 3

A manufacturing company produces widgets at a cost of $12 per unit plus a fixed daily cost of $480. If the company must keep daily production costs below $2400 and produce at least 50 widgets per day to meet demand, what is the range of widgets $ $x$ $ that can be produced daily?

$[50, 160]$
$(50, 160)$
$[50, 160)$ (correct answer)
$(50, 160]$

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 4

Material testing: Solve $12-3s\ge0$ (s = stress units) and give the solution interval.

$[4,\infty)$
$(-\infty,4]$ (correct answer)
$(-\infty,4)$
$(4,\infty)$

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 12 - 3s ≥ 0 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 when dividing by a negative number, often confusing students about sign changes. 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

Budgeting: Solve $4(x-3)\le 28$ for textbook spending; express solution in interval notation.

$(-\infty,10]$ (correct answer)
$(-\infty,10)$
$(10,\infty)$
$[10,\infty)$

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 4(x-3) ≤ 28 requires isolating the variable and determining the solution set, represented as (-∞, 10]. The correct choice, A, accurately reflects this solution by properly applying interval notation standards. A common distractor, C, fails because it reverses the inequality, often confusing students about distribution. 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

Traffic flow: Solve $\tfrac{t}{3}+4\ge6$ and express the timing solution in interval notation.

$[6,\infty)$ (correct answer)
$(-\infty,6]$
$(6,\infty)$
$(-\infty,6)$

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 t/3 + 4 ≥ 6 requires isolating the variable and determining the solution set, represented as [6, ∞). The correct choice, A, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it uses a closed interval in the wrong direction, often confusing students about multiplication in 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 7

A rental car company charges a base fee plus a rate per mile. The total cost $C$ for driving $m$ miles satisfies the inequality $45 + 0.25m \leq C < 45 + 0.25m + 15$ . Which interval notation correctly represents the possible costs for driving exactly 200 miles?

$[95, 110)$ (correct answer)
$[95, 110]$
$(95, 110)$
$(95, 110]$

Explanation: For

m = 200

, substitute into the inequality:

45 + 0.25(200) \leq C < 45 + 0.25(200) + 15

. This becomes

45 + 50 \leq C < 45 + 50 + 15

, so

95 \leq C < 110

. In interval notation, this is

[95, 110)

. The left endpoint is included (closed bracket) due to

\leq

, while the right endpoint is excluded (open parenthesis) due to

<

. Choice B incorrectly includes 110. Choices C and D incorrectly exclude 95.

Question 8

Temperature regulation: Solve $5(T-18)\ge10$ and write acceptable $T$ in interval notation.

$(20,\infty)$
$(-\infty,20]$
$[20,\infty)$ (correct answer)
$(-\infty,20)$

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 - 18) ≥ 10 requires isolating the variable and determining the solution set, represented as [20, ∞). The correct choice, C, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it uses the wrong bracket type, often confusing students about inclusive versus exclusive endpoints. 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

Budgeting: Determine $5x+20\ge 70$ for club dues; give solution set in interval notation.

$[10,\infty)$ (correct answer)
$(10,\infty)$
$(-\infty,10]$
$(-\infty,10)$

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 5x + 20 ≥ 70 requires isolating the variable and determining the solution set, represented as [10, ∞). The correct choice, A, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it uses an open parenthesis instead of a bracket, 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 10

Budgeting: Solve $3-9x<30$ for penalty points; express solution in interval notation.

$(-3,\infty)$ (correct answer)
$(-\infty,-3)$
$(-\infty,-3]$
$[-3,\infty)$

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 3 - 9x < 30 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 includes equality, often confusing students about flipping 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 11

Budgeting: Determine $-7\le x-2$ for account balance; write solution in interval notation.

$(-\infty,-5]$
$(-\infty,-5)$
$[-5,\infty)$ (correct answer)
$(-5,\infty)$

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 -7 ≤ x - 2 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, A, fails because it reverses the direction, often confusing students about adding to both sides. 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

Temperature regulation: Determine the solution set for $2T+7\ge -1$ in interval notation.

$[-4,\infty)$ (correct answer)
$(-\infty,-4]$
$(-4,\infty)$
$(-\infty,-4)$

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 2T + 7 ≥ -1 requires isolating the variable and determining the solution set, represented as [-4, ∞). The correct choice, A, accurately reflects this solution by properly applying interval notation standards. A common distractor, B, fails because it reverses the direction incorrectly, often confusing students about subtracting constants. 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

A temperature monitoring system triggers an alert when the temperature $T$ (in °F) satisfies the condition $T - 72 > 8$ OR $T - 72 < -12$ . For what range of temperatures will the system trigger an alert?

$T < 60$ or $T > 80$ (correct answer)
$T \leq 60$ or $T \geq 80$
$60 < T < 80$
$T < 64$ or $T > 84$

Explanation: We need to solve two separate inequalities. First:

T - 72 > 8

gives

T > 80

. Second:

T - 72 < -12

gives

T < 60

. Since these are connected by OR, the solution is

T < 60

T > 80

. Choice B incorrectly uses

\leq

and

\geq

(the original inequalities are strict). Choice C gives the complement (when the system does NOT trigger). Choice D uses wrong values (64 and 84 instead of 60 and 80).

Question 14

Budgeting: Solve $3x+45\le180$ (x = weekly dining dollars) and write the solution in interval notation.

$( -\infty,45]$ (correct answer)
$[45,\infty)$
$( -\infty,45)$
$[ -\infty,45]$

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 15

Budgeting: Determine the solution set for $0.5x+30<80$ and express it in interval notation.

$(-\infty,100)$ (correct answer)
$(-\infty,100]$
$(100,\infty)$
$[100,\infty)$

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 16

Budgeting: Which interval represents the solution to $-6x\ge18$ for weekly savings $x$ ?

$[-3,\infty)$
$(-\infty,-3)$
$( -\infty,-3]$ (correct answer)
$(-3,\infty)$

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 17

Material testing: Which interval represents the solution to $-\tfrac{s}{2}+1\ge0$ ?

$(-\infty,2)$
$[2,\infty)$
$(-\infty,2]$ (correct answer)
$(2,\infty)$

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 18

Traffic flow: Solve $5-\tfrac{t}{2}<1$ and express the timing solution in interval notation.

$(-\infty,8)$
$(8,\infty)$ (correct answer)
$(-\infty,8]$
$[8,\infty)$

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 19

Material testing: Solve $9\le 3s$ and express the safe $s$ range using interval notation.

$(-\infty,3]$
$(-\infty,3)$
$[3,\infty)$ (correct answer)
$(3,\infty)$

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 20

Budgeting: Determine $\frac{x+9}{3}>1$ for surplus; write solution in interval notation.

$(-\infty,-6)$
$(-\infty,-6]$
$(-6,\infty)$ (correct answer)
$[-6,\infty)$

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.