representation on number line

Correct answer: D. ((-\infty,-7)\cup(2,\infty))

(|2x+5|>9) gives (2x+5<-9) or (2x+5>9), so (x<-7) or (x>2). In exams, modulus with (>) gives open outer parts.

Correct answer: B. ([-2,3])

The common part is from (-2) to (3), and both endpoints are included. In exams, take only the common shaded region for (\cap).

Correct answer: B. ((-9,-1])

The open dot excludes (-9), and the closed dot includes (-1). In exams, use a round bracket for an open endpoint.

Correct answer: D. \(x\ge 6\), closed dot at \(6\) and shading to the right

For \(5x-8\ge 2x+10\), add \(2x\) to both sides to get \(3x-8\ge 10\). Adding \(8\) gives \(3x\ge 18\), so \(x\ge 6\). Hence, mark a closed dot at \(6\), since it is included, and shade to the right. Option A has the correct direction but an open dot, which excludes \(6\). Exam tip: use a closed dot for \(\ge\) or \(\le\), and an open dot for \(>\) or \(<\).

Correct answer: C. (x<-4) or (x\ge3)

The first part is open before (-4), and the second starts from (3) inclusive. In exams, read (\cup) as or.

Correct answer: A. ([-11,7))

Multiplying all parts by (3) gives (-6\le x+5<12), so (-11\le x<7). In exams, apply the same operation to every part of a compound inequality.

Correct answer: C. ({-2,-1,0,1,2,3,4})

Integers greater than (-\frac{5}{2}) start at (-2), and the last integer up to (\frac{13}{3}) is (4). In exams, choose valid integers at fractional boundaries.

Correct answer: D. ((-\infty,-6)\cup[2,\infty))

The given interval includes (-6) and excludes (2), so the complement excludes (-6) and includes (2). In exams, check endpoints carefully while taking complements.

Correct answer: B. ((-5,11))

The direct answer is B: \((-5,11)\). The rule \(|u|<8\) is equivalent to \(-8<u<8\). Put u=x-3: \(-8<x-3<8\). Add 3 to all three parts, giving \(-5<x<11\). The endpoints are open because equality is not allowed: x=-5 gives absolute value 8, and x=11 also gives absolute value 8. Option A, \([-5,11]\), incorrectly includes both endpoints. Option B correctly shows the open interval between them. Option C shows the two outside rays, which would describe values with distance greater than 8. Option D shows outside rays including endpoints, which would describe distance at least 8. The correct idea is that a strict absolute-value inequality gives the inside region. Memory cue: less than means inside and open; greater than means outside.

Correct answer: C. ([-\frac{5}{2},2])

(-9\le4x+1\le9) gives (-10\le4x\le8), so (-\frac{5}{2}\le x\le2). In exams, modulus with (\le) gives a closed interval.

Correct answer: B. ((-2,3))

The common part is from (-2) to (3), and both endpoints are excluded. In exams, take only the overlapping region for (\cap).

Correct answer: A. (x<2), open dot at (2) shaded left

(8-6x-3>-7) gives (5-6x>-7), so (x<2). In exams, reverse the sign when dividing by a negative coefficient.

Correct answer: B. ([-4,5))

The first condition gives (x<5), and the second gives (x\ge-4), so the common solution is ([-4,5)). In exams, and means the common shaded part.

Correct answer: C. ((-\infty,4))

The first part includes (-3), and the second starts after (-3), so no gap remains. In exams, merge connected intervals.

Correct answer: D. (5\le x<14)

The closed dot includes (5), and the open dot excludes (14). In exams, decide (\le) or (<) from the endpoint.

Correct answer: A. ([-3,2]\cup(7,9])

The removed part does not remove (2), but it removes (7). In exams, pay attention to the endpoints removed in set difference.

Correct answer: C. ((-\infty,-8]\cup[8,\infty))

(x^2\ge64) gives (|x|\ge8), so two closed outer rays are obtained. In exams, include boundary points when the sign is (\ge).

Correct answer: C. (7)

The natural numbers are (4,5,6,7,8,9,10), so there are (7) points. In exams, do not count the strict upper boundary.

Correct answer: A. (\emptyset)

No number can be less than or equal to (7) and greater than (7) at the same time. In exams, contradictory conditions have an empty intersection.

Correct answer: B. ((-\infty,9)\cup(9,\infty))

For (x\ne9), the whole number line is taken except (9). In exams, show the removed point with an open circle.

Correct answer: B. ((0,4])

The first inequality gives (-2\le x\le4), and the second gives (x>0), so the common part is ((0,4]). In exams, apply the extra condition as an intersection.

Correct answer: A. ((-12,1])

The first interval includes (-5), and the second starts after (-5), so there is no gap. In exams, merge touching intervals.

Correct answer: C. (11)

(|x+2|\le5) gives (-7\le x\le3), so there are (11) integers. In exams, count endpoints in a closed interval.

Correct answer: B. ((-\infty,-2]\cup(4,10))

In (x\le-2), (-2) is included, and in (4<x<10), both endpoints are excluded. In exams, write or using union.

Correct answer: A. (x\le7), closed dot at (7) shaded left

(11-3x\ge-10) gives (-3x\ge-21), so (x\le7). In exams, reverse the inequality while dividing by a negative.