representation on number line

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

(|x-1|<4) gives (-3<x<5), so integers go from (-2) to (4). In exams, do not include strict boundaries.

Correct answer: A. (x\le\frac{19}{2}), closed dot at (\frac{19}{2}) shaded left

(7-2x\ge -12) gives (-2x\ge -19), so (x\le\frac{19}{2}). In exams, reverse the sign when dividing by a negative.

Correct answer: B. ((-\infty,-10)\cup(-3,\infty))

Because the endpoints are open, (-10) and (-3) are excluded, and two outer parts are taken. In exams, outer shading means two separate rays.

Correct answer: B. (x>4), open dot at (4) shaded right

Multiplying by (6) gives (3x+3<3x-5), which is impossible. In exams, when (x)-terms cancel, compare the constants.

Correct answer: B. ([3,12])

The common part is from (3) to (12), and both endpoints are included. In exams, two included boundaries give a closed interval in the common region.

Correct answer: A. ((-\infty,-4]\cup(0,2])

The removed part does not remove (-4), but it removes (0). In exams, think carefully about the open and closed endpoints of the removed interval.

Correct answer: C. (−∞, 5) ∪ (9, ∞)

The modulus |x − 7| measures the distance of x from 7. The inequality |x − 7| > 2 therefore means that x is more than 2 units away from 7. Splitting the absolute-value inequality gives x − 7 > 2 or x − 7 < −2. These simplify to x > 9 or x < 5. Because the inequality is strict, the boundary values 5 and 9 are excluded; this is why both endpoints are represented by open circles. On the number line, the solution consists of the two outer rays, extending left from 5 and right from 9: (−∞, 5) ∪ (9, ∞). Option A represents the inside interval and option B includes the boundary, so both are wrong. Option D incorrectly includes 5 and 9. Thus option C is correct.

Correct answer: A. ((-2,9))

The two intervals overlap and cover from (-2) to (9) without a gap. In exams, take the entire covered region for a union.

Correct answer: B. ([-7,-5])

The common part is from (-7) to (-5), and both endpoints are included. In exams, choose only the overlapping segment for intersection.

Correct answer: B. (x\le3), closed dot at (3) shaded left

(8-2x\ge3x-7) gives (15\ge5x), so (x\le3). In exams, keep variables on one side and constants on the other.

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

The given parts cover the whole number line except (-1). In exams, identify the missing point separately.

Correct answer: A. ([1,6])

(5\le2x+3\le15) gives (2\le2x\le12), so (1\le x\le6). In exams, a closed compound inequality gives a closed interval.

Correct answer: C. (\emptyset)

The first part excludes (5), and the second starts at (5), so there is no common point. In exams, decide intersection only after checking endpoint inclusion.

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

The closed dot includes (-8), and right shading means (x\ge -8). In exams, read a right ray as the greater side.

Correct answer: B. (x>1), open dot at (1) shaded right

Multiplying by (-2) reverses the sign to (9-x>-8), so (x<17), not (x>17). In exams, reverse the sign when removing a negative denominator.

Correct answer: A. ((-\infty,0]\cup(2,7))

The first part is left including (0), and the second is open between (2) and (7). In exams, connect separate conditions using union.

Correct answer: B. ((-\infty,6)\cup(6,\infty))

Only (6) is removed, so there is an open dot at (6) and shading on both sides. In exams, identify this as (x\ne6).

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

The first interval includes (2), and the second starts after (2), so there is no gap. In exams, merge connected intervals into one.

Correct answer: A. (x>-3), open dot at (-3) shaded right

The direct answer is option A: x>-3, with an open dot at -3 and shading to the right. Solve step by step: 6-4x<2x+24. Subtract 6 from both sides: -4x<2x+18. Subtract 2x: -6x<18. Dividing by -6 reverses the inequality sign, giving x>-3. Because the inequality is strict, -3 is not included, so the dot is open; values greater than -3 lie to the right. Option B has the wrong direction. Option C uses a closed dot and includes -3, which is not allowed. Option D has both the wrong inclusion and the wrong direction. Checking x=0 confirms the result: 6<24 is true, and 0 lies right of -3. Common mistake: always reverse the sign when dividing or multiplying by a negative number.

Correct answer: B. ((4,10))

The common part is greater than (4) and less than (10), so both endpoints are open. In exams, read both inequalities together in an intersection.

Correct answer: A. Smallest (4), greatest (11)

The solution (6\le x+2<14) gives (4\le x<12), so integers run from (4) to (11). In exams, take the integer just before a strict upper boundary.

Correct answer: A. (x\le\frac{32}{7}), closed dot at (\frac{32}{7}) shaded left

Multiplying by (6) gives (10x-8\le3x+24), so (x\le\frac{32}{7}). In exams, multiply by the LCM to clear fractions.

Correct answer: A. ((-5,1])

(4\le7-3x<22) gives (-3\le-3x<15), so (-5<x\le1). In exams, dividing by a negative changes both order and signs.

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

The first integer greater than or equal to (-\frac{11}{3}) is (-3), and the last integer less than (\frac{9}{2}) is (4). In exams, choose valid integers separately at fractional boundaries.

Correct answer: C. ([-6,-1]\cup(3,4))

The removed part does not remove (-1), but it removes (3). In exams, check the open and closed endpoints of the removed interval carefully.