GMAT Math : Graphing a two-step inequality

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Example Question #1 : Graphing A Two Step Inequality

Axes_2

Which of the following inequalities is graphed above?

Possible Answers:

$y> -\frac{5}{4} x + 5$

$y< -\frac{5}{4} x -5$

$y< -\frac{5}{4} x +\frac{25}{4}$

$y> -\frac{5}{4} x -\frac{25}{4}$

$y< -\frac{5}{4} x -\frac{25}{4}$

Correct answer:

$y> -\frac{5}{4} x -\frac{25}{4}$

Explanation:

First, we determine the equation of the boundary line. This line includes points (-1,-5) and (-5,0) , so the slope can be calculated as follows:

$m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{-5-0}{-1- (-5)} =- \frac{5}{4}$

We can find the slope-intercept form of the line by substituting $m =- \frac{5}{4}, x = -5,y= 0$

in the following equation:

y = mx + b

$0 = -\frac{5}{4} (-5) + b$

$0 = \frac{25}{4} + b$

$b = -\frac{25}{4}$

The equation of the boundary line is $y= -\frac{5}{4} x -\frac{25}{4}$ .

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 (0,0) :

_____ $-\frac{5}{4} x -\frac{25}{4}$

_____ $-\frac{5}{4} \cdot 0 -\frac{25}{4}$

_____ $-\frac{25}{4}$

0 is greater than $-\frac{25}{4}$ so the correct symbol is .

The correct choice is $y> -\frac{5}{4} x -\frac{25}{4}$ .

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Example Question #2 : Graphing A Two Step Inequality

Choose the inequality depicted by the graph:

Possible Answers:

$y\geq2x+3$

y<2x+3

y>2x+3

$y\leq2x+3$

Correct answer:

$y\geq2x+3$

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.

0<2(0)+3=3

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"

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Example Question #3 : Graphing A Two Step Inequality

Capture1

Which of the following inequalities is graphed above?

Possible Answers:

$y\leq 2x+5$

y> 2x+5

y<2x+5

None of the above.

$y\geq 2x+5$

Correct answer:

y<2x+5

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 (0,5) and (-5,-5) , so the slope of the line is

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-5 -5}{-5-0}=\frac{-10}{-5}=2$ .

We can now find the -intercept form of the line by substituting m=2 and the point (x,y)= (0,5) into the slope-intercept equation y=mx+b and solving for :

(5)=(2)(0)+b

5=b

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 (0,0) since it's the simplest to substitute:

_____ 2x+5

_____ 2(0)+5

_____

0<5 , so the correct symbol is : y<2x+5

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Example Question #4 : Graphing A Two Step Inequality

Capture2

Which of the following inequalities is graphed above?

Possible Answers:

$y\leq \frac{1}{2}x+3$

$y> \frac{1}{2}x+3$

$y\geq \frac{1}{2}x+3$

$y< \frac{1}{2}x+3$

None of the above.

Correct answer:

$y\geq \frac{1}{2}x+3$

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 (0,3) and (-6,0) , so the slope of the line is

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{0 -3}{-6-0}=\frac{-3}{-6}=\frac{1}{2}$ .

We can now find the -intercept form of the line by substituting $m=\frac{1}{2}$ and the point (x,y)= (0,3) into the slope-intercept equation y=mx+b and solving for :

$(3)=\left(\frac{1}{2}\right)(0)+b$

3=b

The equation of the boundary line is therefore $y=\frac{1}{2}x+3$ . 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 $\leq$ or a $\geq$ .

In order to determine which one, we can test a point in the solution set; let's test (0,5) :

_____ $\frac{1}{2}x+3$

_____ $\frac{1}{2}(0)+3$

_____

5>3 , so the correct symbol is $\geq$ : $y\geq \frac{1}{2}x+3$

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Example Question #5 : Graphing A Two Step Inequality

Capture3

Which of the following inequalities is graphed above?

Possible Answers:

None of the above.

$y\geq 4x+2$

y>4x+2

$y\leq 4x+2$

y< 4x+2

Correct answer:

y>4x+2

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 $\left(-\frac{1}{2},0\right)$ and (0,2) , so the slope of the line is

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{2 -0}{0-(-\frac{1}{2})}=\frac{2}{\frac{1}{2}}=4$ .

We can now find the -intercept form of the line by substituting m=4 and the point (x,y)= (0,2) into the slope-intercept equation y=mx+b and solving for :

(2)=(4)(0)+b

2=b

The equation of the boundary line is therefore y=4x+2 . 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 (-1,1) :

_____ 4x+2

_____ 4(-1)+2

_____

-1>-2 , so the correct symbol is : y> 4x+2

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GMAT Math : Graphing a two-step inequality

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Example Question #1 : Graphing A Two Step Inequality

Example Question #2 : Graphing A Two Step Inequality

Example Question #3 : Graphing A Two Step Inequality

Example Question #4 : Graphing A Two Step Inequality

Example Question #5 : Graphing A Two Step Inequality

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