representation on number line

Correct answer: C. When the inequality has (<) or (>)

In a strict inequality the boundary value is not included so the endpoint is open. In exams connect (<) and (>) with open dots.

Correct answer: B. (x<-4)

The value (-4) is excluded and all smaller values are included so (x<-4). In exams read an interval with (-\infty) as a left ray.

Correct answer: C. ((6,\infty))

The open dot excludes (6) and right shading shows values greater than (6). In exams never use a square bracket with (\infty).

Correct answer: B. As (x\ge 2) with closed dot and right ray

The solution gives (-4x\le -8) and dividing by a negative gives (x\ge 2). In exams forgetting sign reversal is the most common error.

Correct answer: A. Open dot at (3) and closed dot at (8)

In ((3,8]) the value (3) is excluded and (8) is included. In exams convert interval notation directly into endpoint dots.

Correct answer: B. (2\le x<8)

Multiplying the whole inequality by (2) gives (2\le x<8). In exams the sign does not change when multiplying by a positive number.

Correct answer: C. ([-2,5))

The value (-2) is included and (5) is excluded so ([-2,5)) is correct. In exams read the left and right endpoints separately.

Correct answer: D. Closed dot at (0) and ray right

In (x\ge 0) the value (0) is included and greater values are to the right. In exams treat (0) as a boundary value too.

Correct answer: B. Only a closed dot at (2)

This inequality is satisfied only by (x=2) so one closed dot is drawn. In exams treat a closed interval with equal endpoints as a single point.

Correct answer: C. No real number

In ((4,4)) both endpoints are open and no number remains between them. In exams an interval with equal open endpoints is empty.

Correct answer: B. ([-6,-1))

There is equality at (-6) so use a square bracket and no equality at (-1) so use a round bracket. In exams practice converting signs into brackets.

Correct answer: C. (x<-\frac{1}{3})

The open dot excludes the boundary and the left direction shows smaller values. In exams pay special attention to the direction for a negative fraction.

Correct answer: B. Closed dot at (0) and ray left

The interval ((-\infty,0]) includes (0) and all smaller values. In exams remember ((-\infty,0]) as (x\le 0).

Correct answer: C. (x\le 6)

Simplifying gives (x\le 6) so there is a closed dot at (6) and a left ray. In exams bring variable terms to one side before representing.

Correct answer: D. ((0,5))

The two conditions together give (0<x<5) and both endpoints are open. In exams and means the common shaded part.

Correct answer: B. ((-\infty,-2]\cup(3,\infty))

For or take the union of both separate regions. In exams keep a closed endpoint for (\le -2) and an open endpoint for (>3).

Correct answer: A. (-4<x<1)

Both open endpoints indicate strict inequalities so (-4<x<1) is correct. In exams write open endpoints using (<).

Correct answer: B. As (x\ge 3) with right ray

The inequality (3\le x) can also be written as (x\ge 3). In exams writing the variable on the left makes the direction easier to read.

Correct answer: B. (0) closed and (\frac{9}{4}) open

There is equality at (0) so it is closed and no equality at (\frac{9}{4}) so it is open. In exams draw mixed endpoints carefully.

Correct answer: C. \(x>5\)

In \(x>5\), the boundary value 5 is not included, so an open circle is drawn at 5 and the ray extends rightward. Check: substituting \(x=5\) makes the statement false. Exam tip: \(\geq\) and \(\leq\) use a closed circle.

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

The whole real line is written as ((-\infty,\infty)) and square brackets are not used with infinity. In exams do not treat infinity as an includable endpoint.

Correct answer: C. ((-9,\infty))

The open dot excludes (-9) and the right ray takes values greater than it. In exams write (>) as ((a,\infty)).

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

The two parts are separate and both (1) and (4) are included. In exams show both rays separately for a union.

Correct answer: B. (-2<x\le 4)

Dividing by (-3) reverses both inequality signs and gives (-2<x\le 4). In exams keep the order correct after negative division.

Correct answer: C. (x\ge 11)

The closed dot includes (11) and the right direction shows greater values. In exams choose the equality sign with a closed dot.