# GMAT Math : Graphing a two-step inequality

## Example Questions

### Example Question #235 : Coordinate Geometry

Which of the following inequalities is graphed above?

Explanation:

First, we determine the equation of the boundary line. This line includes points  and  , so the slope can be calculated as follows:

We can find the slope-intercept form of the line by substituting

in the following equation:

The equation of the boundary line is .

The boundary is excluded, as is indicated by the line being dashed, so the equality symbol is replaced by either  or . To find out which one, we can test a point in the solution set - for ease, we will choose :

_____

_____

_____

0 is greater than  so the correct symbol is

The correct choice is .

### Example Question #1 : Graphing A Two Step Inequality

Choose the inequality depicted by the graph:

Explanation:

First, consider the characteristics of the line. The slope is equal to 2 and the y-intercept is equal to 3. Because the line is solid, that indicates that the inequality is "greater than or equal to" or "less than or equal to". Finally, choose a point to determine the direction of the shading. The origin (0,0) is usually a good choice unless it falls on the line. If the chosen point makes the statement true, it must be included in the shaded region. If it is false, it must not.

Because 0 is less than 3 and the origin is not included in the shaded region, the correct answer must include "greater than or equal to"

### Example Question #2 : Graphing A Two Step Inequality

Which of the following inequalities is graphed above?

None of the above.

Explanation:

In order to graph the inequality pictured above, we must first find the equation of its boundary line. Based on the image, we see that the line includes the points  and , so the slope of the line is

.

We can now find the -intercept form of the line by substituting  and the point   into the slope-intercept equation  and solving for :

The equation of the boundary line is therefore . Since we see that the boundary line is dashed, we know that the values on the line are excluded from the inequality, so the  sign will be replaced by a  or a .

In order to determine which one, we can test a point in the solution set; let's test  since it's the simplest to substitute:

_____

_____

_____

, so the correct symbol is

### Example Question #3 : Graphing A Two Step Inequality

Which of the following inequalities is graphed above?

None of the above.

Explanation:

In order to graph the inequality pictured above, we must first find the equation of its boundary line. Based on the image, we see that the line includes the points  and , so the slope of the line is

.

We can now find the -intercept form of the line by substituting  and the point   into the slope-intercept equation  and solving for :

The equation of the boundary line is therefore . Since we see that the boundary line is solid, we know that the values on the line are included in the inequality, so the  sign will be replaced by a  or a .

In order to determine which one, we can test a point in the solution set; let's test  :

_____

_____

_____

, so the correct symbol is

### Example Question #4 : Graphing A Two Step Inequality

Which of the following inequalities is graphed above?

None of the above.

Explanation:

In order to graph the inequality pictured above, we must first find the equation of its boundary line. Based on the image, we see that the line includes the points  and , so the slope of the line is

.

We can now find the -intercept form of the line by substituting  and the point   into the slope-intercept equation  and solving for :

The equation of the boundary line is therefore . Since we see that the boundary line is dashed, we know that the values on the line are excluded from the inequality, so the  sign will be replaced by a  or a .

In order to determine which one, we can test a point in the solution set; let's test  :

_____

_____

_____

, so the correct symbol is