representation on number line

Correct answer: B. ((-\infty,-1]\cup[4,\infty))

((-1,4)) does not include (-1) and (4), so the complement includes both. Hence the outside regions have closed endpoints.

Correct answer: B. Closed circle at (1) and right side

Solving gives (5-2x\le 3), then (-2x\le -2), and (x\ge 1). The boundary is included, so (1) is closed.

Correct answer: A. ([-5,4])

The first interval includes (0), and the second starts after (0). So no gap remains and the result is ([-5,4]).

Correct answer: A. A closed point at 3 and shading to the left

In \(x\le3\), the endpoint 3 is included, so it is shown by a closed point. Numbers smaller than 3 lie to the left, so that side is shaded. Exam tip: use a closed point for \(\le\) or \(\ge\), not an open point.

Correct answer: A. {1}

The governing concept is intersection, which consists only of numbers common to both sets. The interval (-∞,1] contains every real number less than or equal to 1, so it includes 1. The interval [1,∞) contains every real number greater than or equal to 1, and it also includes 1. A number smaller than 1 fails to belong to the second interval, while a number greater than 1 fails to belong to the first. Consequently, the only common number is 1. The intersection is therefore the singleton set {1}, represented by one closed point at 1 on the number line. Option A is correct; the other choices either include too many points or incorrectly claim that no point is common.

Correct answer: A. Open circle at 2 and shading to the right

Solving the inequality gives 4x+1>9, so 4x>8 and x>2. The boundary point 2 is not included because the inequality is strict; therefore, an open circle is placed at 2. All values greater than 2 lie to the right, so the right side is shaded. A closed circle would represent an inclusive inequality such as x≥2 or x≤2. Exam tip: use an open circle for > or < and a closed circle for ≥ or ≤.

Correct answer: C. ([-3,\infty))

Solving gives (-5x\le 15) and then (x\ge -3). The inequality sign reverses when dividing by a negative.

Correct answer: B. ((-2,4))

The common part is greater than (-2) and less than (4). Both endpoints are excluded in this intersection.

Correct answer: A. \(( -\infty,5)\)

Combine the intervals: \((-\infty,-1)\cup[-1,5)\) includes all \(x\) less than \(5\) and also includes \(-1\) because \(-1\) belongs to \([-1,5)\). Since \(5\) is not included (\(5\) has a parenthesis), the simplified interval is \(( -\infty,5)\).

Correct answer: B. (-∞,-4) ∪ [2,∞)

The governing concept is the complement of an interval in the set of real numbers. In [-4,2), the point -4 is included because of the closed bracket, while 2 is excluded because of the open parenthesis. The complement must contain all real numbers outside this interval. Numbers less than -4 are included, but -4 itself is not, giving (-∞,-4). Numbers greater than or equal to 2 are included, because 2 was excluded from the original interval, giving [2,∞). Therefore the complement is (-∞,-4) ∪ [2,∞), so option B is correct. Option A reverses both endpoint memberships, C gives the original interval, and D wrongly excludes 2.

Correct answer: B. ([4,9))

Solving gives (5\le 2x-3<15) and then (4\le x<9). Therefore (4) is included but (9) is excluded.

Correct answer: B. 7

In \(-4\le x<3\), \(-4\) is included, but \(3\) is not included. Thus, the integers are \(-4,-3,-2,-1,0,1,2\), giving a total of \(7\) points. Counting \(3\) would be incorrect because the inequality is \(x<3\). Exam tip: Include an endpoint with \(\le\) and exclude an endpoint with \(<\).

Correct answer: C. Open circle at \(4\), shade to the right

\(x>4\) means values greater than \(4\), so \(4\) itself is not included. Hence we use an open circle at \(4\) and shade to the right.

Correct answer: B. Closed circle at (-3) and line to the left

In (\le), the boundary number is included and smaller numbers lie to the left. In exams, a closed circle means included.

Correct answer: A. Open circle at (2) closed circle at (7) and shaded between

The direct answer is option A. The interval \((2,7]\) has two different bracket meanings. A round or open bracket at 2 means 2 is excluded, so the graph must have an open circle at 2. A square or closed bracket at 7 means 7 is included, so the graph must have a filled circle at 7. Every real number between them is included, so shade the segment between the two points. Thus the representation is an open circle at 2, a closed circle at 7, and shading between them. Option A matches exactly. Option B reverses both endpoint types. Option C incorrectly excludes 7 as well as 2. Option D incorrectly includes 2 and also includes 7. In inequality form, the same interval is \(2<x\le7\). Remember that a parenthesis means “not included” and a square bracket means “included”; the bracket closest to each number tells its status.

Correct answer: B. ([6,\infty))

A closed circle includes (6), and numbers to the right are greater. In exams, always use an open bracket with (\infty).

Correct answer: A. (x<-2)

An open circle excludes the boundary and the left side has smaller numbers. In exams, connect left shading with (<).

Correct answer: A. When the endpoint is included in the solution, such as \(x \le a\) or \(x \ge a\)

A filled circle shows that the endpoint itself belongs to the solution set. Hence, for \(x \le a\) and \(x \ge a\), a filled circle is marked at \(a\). In \(x < a\) or \(x > a\), the endpoint is excluded, so an open circle is used. Exam tip: associate \(\le\) and \(\ge\) with a filled circle, and \(<\) and \(>\) with an open circle.

Correct answer: D. (4)

In ([0,4)), (0) is included but (4) is not included. In exams, exclude the endpoint with an open bracket.

Correct answer: C. 5

The interval \((-5,1)\) excludes \(-5\) and \(1\). The integers are \(-4,-3,-2,-1,0\). That is \(5\) integers.

Correct answer: A. Open circle at 4 and shading to the left

Given \(2x+1<9\), subtracting 1 from both sides gives \(2x<8\). Dividing by 2 gives \(x<4\). Thus, 4 is not included, so the number line has an open circle at 4 and is shaded to the left for all values less than 4. Option B represents \(x\le4\), incorrectly including 4. Exam tip: use an open circle for \(<\) or \(>\), and a closed circle for \(\le\) or \(\ge\).

Correct answer: C. ([-3,5])

Dividing all parts by (2) gives (-3\le x\le 5). In exams, apply the same operation to every part of a compound inequality.

Correct answer: B. ((2,8])

Multiplying all parts by (2) gives (2<x\le 8). In exams, check carefully whether each endpoint is included.

Correct answer: C. (6)

The integers are (-2,-1,0,1,2,3), so the total is (6). In exams, count both endpoints in a closed interval.

Correct answer: B. ((0,9))

Both circles are open, so both endpoints are excluded. In exams, write an open circle using an open bracket.