representation on number line

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

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.

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

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.

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

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

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

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.

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.

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

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.

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.

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

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.

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

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

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

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.

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.

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.

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.

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

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

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

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