Solve graphically the inequality \[ x ≥ 2 \]

1) Graph the corresponding equation \( x = 2 \); this will split the plane into two regions. One of the region represents the solution set.

2) Select a point situated in any of the two regions obtained and test the inequality. If the point selected is a solution, then all the region is the solution set. If the selected point is not a solution, then the other (second) region represents the solution set.

3) Test: In this example, let us for example select the point with coordinates (3 , 2) which is in the region to the right of the line x = 2. If you substitute x in the inequality \( x ≥ 2 \) by 3 it becomes \( 3 ≥ 2 \) which is a true statement and therefore (3 , 2) is a solution. Hence, we can conclude that the region to the right of the vertical line x = 2 is a solution set including the line itself which is shown as a solid line because of the inequality symbol \( ≥ \) contains the \( = \) symbol. The solution set is represented by the blue hash region in the graph below including the line x = 2.

The only correct answer is: –6x + 15 < 10 – 5x.

–3(2x – 5) < 5(2 – x) simplifies to –6x + 15 < 10 – 5x 

now, if you continue to reduce the inequality:

–6x + 15 < 10 – 5x    subtract 15 from both sides

–6x <  –5 – 5x           add 5x to both sides

–x <  –5                     multiply each side by -1

x > 5                          flip the inequality sign since you multiplied by a negative number