representation on number line

Correct answer: A. ((-\infty,-3]\cup[4,\infty))

(|2x-1|\ge 7) gives (2x-1\le -7) or (2x-1\ge 7), so (x\le -3) or (x\ge 4). In exams, absolute value with (\ge) forms closed outer rays.

Correct answer: A. (x\ge 3), closed dot at (3) shaded right

(4-3x\le -5) gives (-3x\le -9), so (x\ge 3). In exams, reverse the inequality when dividing by a negative coefficient.

Correct answer: C. ((-6,-2)\cup[0,1])

Removing ([-2,0)) also removes (-2) but keeps (0). In exams, check endpoint inclusion carefully in set difference.

Correct answer: A. ((-5,2])

The open dot excludes (-5), and the closed dot includes (2). In exams, write an open dot with a round bracket and a closed dot with a square bracket.

Correct answer: B. (x\le -3), closed dot at (-3) shaded left

From (-4x\ge 12), dividing by a negative gives (x\le -3). In exams, reverse the inequality sign when dividing by a negative.

Correct answer: C. (x\le -6) or (x>4)

The first part is to the left including (-6), and the second part is to the right after (4). In exams, read (\cup) as or.

Correct answer: A. ((14,34])

Multiplying all parts by (4) gives (12<x-2\le32), so (14<x\le34). In exams, apply the same operation to every part of a compound inequality.

Correct answer: B. ({-7,-6,-5,-4,-3,-2,-1,0,1,2})

(-7) is included, and integers less than (\frac{5}{2}) go up to (2). In exams, with an integer restriction, mark separate points instead of a continuous line.

Correct answer: B. ((-\infty,-3]\cup(5,\infty))

The given interval excludes (-3) and includes (5), so its complement includes (-3) and excludes (5). In exams, check endpoint inclusion carefully while taking complements.

Correct answer: A. \( (-10,2) \)

\( |x+4|<6 \) means the distance of \(x\) from \(-4\) is less than \(6\). So \(-6<x+4<6\). Subtract \(4\) throughout: \(-10<x<2\). Since it is \(<6\) (strict inequality), the endpoints \(-10\) and \(2\) are not included, so the interval is \( (-10,2) \).

Correct answer: B. ([- \frac{8}{3},4]) / ([-\frac{8}{3},4])

(-10\le3x-2\le10) gives (-8\le3x\le12), so (-\frac{8}{3}\le x\le4). In exams, keep both endpoints closed for (\le).

Correct answer: B. ((-4,-1))

The common part is between (-4) and (-1), and both endpoints are excluded. In exams, take only the commonly shaded part for (\cap).

Correct answer: A. (x<\frac{11}{3}), open dot at (\frac{11}{3}) shaded left

(5-6x+2>-15) gives (7-6x>-15), so (x<\frac{11}{3}). In exams, expand brackets and reverse the sign for a negative coefficient.

Correct answer: A. ([-4,6))

The first condition gives (x<6), and the second gives (x\ge -4), so the common solution is ([-4,6)). In exams, take the common region for both conditions.

Correct answer: C. ((-\infty,7))

The first part is before (0), and the second includes (0) up to (7), so no gap remains. In exams, combine connected parts into one interval.

Correct answer: D. (-1\le x<9)

The closed dot includes (-1), and the open dot excludes (9). In exams, decide (\le) or (<) from the type of endpoint.

Correct answer: A. ([-2,1]\cup(6,8])

The removed part does not remove (1), but it removes (6). In exams, check the endpoints of the removed interval carefully.

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

(x^2\le25) means (|x|\le5), so (-5\le x\le5). In exams, take the closed middle part when the sign is (\le).

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

(x^2>49) gives (|x|>7), so two open outer regions are obtained. In exams, do not include boundary points for strict (>).

Correct answer: B. (6)

The natural numbers are (4,5,6,7,8,9), so there are (6) points. In exams, count only the separate valid natural number values.

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

Together, the two parts cover every real number. In exams, the union of complementary rays gives the whole number line.

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

(-6\le5-x<12) gives (-11\le -x<7), so (-7<x\le11). In exams, reverse order and signs when changing (-x) to (x).

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

For (x\ne -4), the whole number line is taken except (-4). In exams, show the removed point as an open dot.

Correct answer: A. ([-6,\infty))

The first part gives (x\le6) and the second gives (x>-3), so the correct common solution is ((-3,6]). In exams, split a compound inequality into two conditions and check carefully.

Correct answer: A. ((-8,3])

The first interval includes (-2), and the second starts after (-2), so no gap remains. In exams, merge connected intervals.