representation on number line

Correct answer: C. ((-\infty,0]\cup[12,\infty))

Because the endpoints are closed, (0) and (12) are included, and two outer rays are taken. In exams, read outer shading as a union of two intervals.

Correct answer: B. All real numbers

Multiplying by (6) gives (2x-8<2x+5), that is (-8<5), which is always true. In exams, when (x)-terms cancel, check the truth of the remaining statement.

Correct answer: B. ([6,15])

The common part is from (6) to (15), and both endpoints are included. In exams, keep closed boundaries in the intersection.

Correct answer: A. ((-\infty,-2)\cup[1,4])

The removed part removes (-2) but does not remove (1). In exams, check the remaining endpoints separately in set difference.

Correct answer: C. (−∞, −9) ∪ (−3, ∞)

The expression |x + 6| represents the distance of x from −6. Requiring this distance to be greater than 3 means that x must lie more than 3 units to the left or right of −6. Algebraically, |x + 6| > 3 becomes x + 6 > 3 or x + 6 < −3. Therefore x > −3 or x < −9. Since the sign is strict, neither boundary −9 nor −3 is included, so both endpoints are open. The number-line solution is consequently the union of the two exterior intervals (−∞, −9) ∪ (−3, ∞). Option A gives the interior interval and reverses the region, option B includes boundaries, and option D also incorrectly uses closed endpoints. Thus option C is the only correct representation.

Correct answer: A. ((-7,6))

The intervals overlap and cover from (-7) to (6) without a gap. In exams, take the entire covered region for a union.

Correct answer: B. ([-9,-6])

The common part is from (-9) to (-6), and both endpoints are included. In exams, choose only the overlapping segment in an intersection.

Correct answer: A. (x\ge -1), closed dot at (-1) shaded right

(15-3x\le2x+20) gives (-5\le5x), so (x\ge-1). In exams, write the final inequality in the standard direction.

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

The given parts cover the whole number line except (2). In exams, identify the missing point while simplifying intervals.

Correct answer: B. (9)

Dividing gives (-4<x\le5), so integers from (-3) to (5) total (9). In exams, do not count an open lower boundary.

Correct answer: A. ([3,7])

(8\le3x-1\le20) gives (9\le3x\le21), so (3\le x\le7). In exams, a closed compound inequality gives a closed interval.

Correct answer: C. (\emptyset)

The first part goes up to (-2), and the second starts after (-2), so there is no common point. In exams, decide intersection after checking endpoint inclusion.

Correct answer: D. ((-\infty,13))

The open dot excludes (13), and left shading means (x<13). In exams, read a left ray as the less than side.

Correct answer: B. (x\ge-2), closed dot at (-2) shaded right

Multiplying by (-4) reverses the sign to (x-10\ge-12), so (x\ge-2). In exams, reverse the sign when removing a negative denominator.

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

Only (-5) is removed, so both sides are shaded with an open point at (-5). In exams, identify it as (x\ne-5).

Correct answer: A. ([-8,2))

The first interval includes (-3), and the second starts after (-3), so no gap remains. In exams, change connected intervals into one interval.

Correct answer: A. (x>-2), open dot at (-2) shaded right

(7-5x<2x+21) gives (-14<7x), so (x>-2). In exams, match the final result with the number line direction.

Correct answer: B. ((6,11))

The common part is greater than (6) and less than (11), so both endpoints are open. In exams, read both conditions together in an intersection.

Correct answer: C. Smallest (13), greatest (26)

(12\le x-1<27) gives (13\le x<28), so integers go from (13) to (27). In exams, take the integer before the strict upper boundary.

Correct answer: A. (x\le19), closed dot at (19) shaded left

Multiplying by (6) gives (4x+14\le3x+33), so (x\le19). In exams, multiply by the LCM to clear fractions.

Correct answer: C. ((-16,-\frac{5}{2}])

(18\le13-2x<45) gives (5\le-2x<32), so (-16<x\le-\frac{5}{2}). In exams, reverse order and signs when dividing by a negative.

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

The first integer greater than (-\frac{17}{4}) is (-4), and the last integer up to (\frac{7}{3}) is (2). In exams, choose valid integers carefully at fractional boundaries.

Correct answer: D. ([-5,-3]\cup(2,4])

The first common part is ([-5,-3]), and the second is ((2,4]). In exams, find the intersection of each union part separately.

Correct answer: B. ((-\infty,-\frac{7}{3}]\cup[5,\infty))

(|3x-4|\ge11) gives (3x-4\le-11) or (3x-4\ge11), so (x\le-\frac{7}{3}) or (x\ge5). In exams, modulus with (\ge) gives closed outer rays.

Correct answer: A. ((-\infty,-2)\cup[4,6])

The removed part removes (-2) but does not remove (4). In exams, check separately which endpoint remains in set difference.