representation on number line

Correct answer: A. −2 ≤ x < 5

The square bracket at −2 means that endpoint is included, so x can equal −2. The round bracket at 5 means that endpoint is excluded. Thus the interval is represented by −2 ≤ x < 5, making option A correct.

Correct answer: A. 3x − 6 ≤ 0

For option A, adding 6 gives 3x ≤ 6, and dividing by 3 gives x ≤ 2. This is exactly the interval (−∞, 2], which includes 2.

Correct answer: B. x ≤ −6

This interval contains all real numbers less than −6 and includes −6 itself, as shown by the square bracket. Therefore, it is written as x ≤ −6.

Correct answer: B. \(2<x<7\)

In the open interval \((2,7)\), neither endpoint 2 nor endpoint 7 is included, so \(2<x<7\) is correct. Option A includes both endpoints, option C includes 2, and option D includes 7. Exam tip: round brackets indicate that the corresponding endpoint is excluded.

Correct answer: A. \(\left(-\infty,-4\right]\cup\left(1,\infty\right)\)

The interval for \(x\leq -4\) is \(\left(-\infty,-4\right]\) because \(-4\) is included. The interval for \(x>1\) is \(\left(1,\infty\right)\) because \(1\) is excluded. The word “or” means the union of these two intervals, giving \(\left(-\infty,-4\right]\cup\left(1,\infty\right)\). Option B reverses the inclusion status of both finite endpoints. Exam tip: use square brackets for \(\leq\) or \(\geq\), and parentheses for strict inequalities.

Correct answer: C. \(x=6\)

In the interval \([0,6)\), 0 is included because it has a square bracket, whereas 6 is excluded because it has a round bracket. Both 5 and \(\frac{1}{2}\) lie between 0 and 6, so they are included. Exam tip: a square bracket means included, while a round bracket means excluded.

Correct answer: A. Closed dot at (-3) shaded left and open dot at (1) shaded right

(x\le -3) includes (-3), while (x>1) does not include (1). In exams, remember closed dot for (\le) and open dot for (>).

Correct answer: B. (-4<x\le 7)

A round bracket means open endpoint and a square bracket means closed endpoint. In exams, use this rule while converting interval notation to inequality.

Correct answer: B. (x>2), shaded right of (2)

From (-3x<-6), dividing by a negative reverses the sign. In exams, reverse the inequality when dividing by a negative.

Correct answer: C. \(-1<x\le 4\)

A single linear inequality in one variable, such as \(ax+b<0\) with \(a\ne0\), gives a ray of the form \(x<c\), \(x\le c\), \(x>c\), or \(x\ge c\). Option C, \(-1<x\le4\), is a bounded interval with two finite endpoints. It requires both conditions \(x>-1\) and \(x\le4\) simultaneously. Options A, B, and D are each single-boundary rays. Exam tip: A solution set with two finite endpoints usually requires more than one inequality.

Correct answer: B. ((3,8))

Both endpoints are open, so (3) and (8) are not included. In exams, write open circles with round brackets.

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

(|x-2|\le 5) gives (-5\le x-2\le 5), hence (-3\le x\le 7). In exams, absolute value with (\le) gives a closed middle interval.

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

(|x+1|>4) gives two outer regions, and endpoints are not included. In exams, absolute value with (>) is shaded outward.

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

Multiplying by (2) gives (x-1\le 6), so (x\le 7). In exams, multiplying by a positive number does not reverse the sign.

Correct answer: B. (x\le -1), closed dot at (-1) shaded left

Solving gives (5-x\ge 6), then (-x\ge 1), so (x\le -1). In exams, remember to reverse the sign when removing (-x).

Correct answer: A. ([-4,0])

In intersection, the common part of both intervals is ([-4,0]). In exams, look for the overlapping region for (\cap).

Correct answer: B. (-2<x\le 6)

The open dot excludes (-2), and the closed dot includes (6). In exams, read endpoint symbols carefully.

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

(4-2x\le 10) gives (-2x\le 6), so (x\ge -3). In exams, reverse the sign when dividing by a negative.

Correct answer: A. (x<-1) or (4\le x\le 9)

The first part is before (-1), and the second part is closed from (4) to (9). In exams, read separate intervals separately.

Correct answer: A. ([-1,3))

The direct answer is option A: the interval is [-1,3). Start with -2≤3x+1<10. Subtract 1 from all three parts: -3≤3x<9. Divide all parts by positive 3, so the inequality signs do not change: -1≤x<3. The left endpoint is included because equality is allowed at -1, so it is shown with a closed point or square bracket. The right endpoint is excluded because the sign is <, so it is shown with an open point or round bracket. Option A matches exactly. Option B reverses both endpoint inclusions and is wrong. Option C uses the original bounds -2 and 10 without solving for x. Option D describes outside the interval and also uses incorrect endpoint directions. Exam cue: for a positive divisor, keep the signs; use square brackets for ≤ and round brackets for <.

Correct answer: B. (3)

(2x<7) gives (x<\frac{7}{2}), so the greatest integer is (3). In exams, choose the integer just before the boundary.

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

(6\le 2x\le 14) gives (3\le x\le 7). In exams, if both signs are inclusive, the final interval stays closed.

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

The complement of ([2,6)) excludes (2) but includes (6). In exams, check endpoints carefully when taking complements.

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

(x^2\ge 16) means (|x|\ge 4), giving two closed outer regions. In exams, include (-4) and (4) for (\ge).

Correct answer: B. (x\ne 2)

The direct answer is option B, x≠2. The union (-∞,2)∪(2,∞) contains every real number less than 2 and every real number greater than 2. The number 2 itself is excluded because neither interval includes its endpoint. That is exactly the meaning of x not equal to 2. Option A, x=2, represents only the single point 2, which is the opposite of this interval. Option C, x≤2, includes all numbers below 2 and also includes 2, but it excludes numbers above 2. Option D, x≥2, includes 2 and numbers above it, but excludes numbers below 2. The open-looking gap at 2 is the important clue. Memory cue: two rays extending in both directions with only one point missing mean ‘all real numbers except that point’.