representation on number line

The first part is open up to (2), and the second starts at (2) included, so no gap remains. In exams, merge connected intervals.

Both intervals exclude (4), so (4) is not included. In exams, watch for missing open points even in a union.

(-1) is included and the line goes right, so (x\ge -1). In exams, read ([ ) as including equality.

Multiplying both sides by (3) gives (x-1>6), so (x>7). In exams, find the boundary first and then choose circle and direction.

The sign (\ge) includes equality so the dot is closed. In exams always check the direction toward greater values.

A round bracket endpoint is not included and a square bracket endpoint is included. In exams remember (() means open and ([) means closed.

The open dot shows that (3.5) is not included and the left direction shows smaller values. In exams write a strict inequality for an open dot.

Solving gives (x\ge 5) so (5) is included and the ray goes right. In exams simplify the inequality first then draw the number line.

Open endpoints mean both (0) and (6) are not included. In exams write ((0,6)) for two open dots.

The sign (>) does not include equality so the dot is open. In exams draw the ray to the right for greater values.

The closed dot includes (-1) and the left direction shows smaller values. In exams use (\le) or (\ge) with a closed dot.

Solving gives (x<3) so (3) is not included. In exams first find the boundary value then decide open or closed dot.

Both open dots show that (0) and (4) are not included. In exams use round brackets for two open endpoints.

The closed dot includes (8) and the right direction shows greater values. In exams choose (>) or (\ge) for a right ray.

The value (-5) is included and (2) is excluded so ([-5,2)) is correct. In exams use a square bracket at the equality endpoint.

The sign (\le) includes equality and smaller values lie to the left. In exams place a fractional boundary like any ordinary number.

The open dot excludes (-3) and the closed dot includes (6). In exams write the shaded region as a compound inequality.

Solving gives (-2x\ge -4) and dividing by a negative changes the sign to (x\le 2). In exams reverse the inequality sign when dividing by a negative.

The interval ((-\infty,9]) includes (9) and all smaller values. In exams the ray toward (-\infty) goes left.

The inequality (x\ne 4) includes all real numbers except (4). In exams show the removed point as an open dot.

The round bracket excludes (-7) and the square bracket includes (-2). In exams the endpoint rule stays the same for negative numbers.

The open dot excludes (1) and the right direction shows greater values. In exams infinity endpoints always use round brackets.

The interval ([-1,3]) has square brackets at both ends so both values are included. In exams draw closed dots for both endpoints of a closed interval.

Both closed dots show that (-8) and (2) are included. In exams connect closed dots with (\le) signs on both sides.

Because of (\ge) the boundary value is included and greater values are to the right. In exams place the negative fraction correctly on the number line.