Graphing a two-step inequality - GMAT Quantitative Reasoning

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Question

Which of the following inequalities is graphed above?

Answer

First, we determine the equation of the boundary line. This line includes points and , 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:

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

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

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

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 :

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

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

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

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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Question

Choose the inequality depicted by the graph:

Answer

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"

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Question

Which of the following inequalities is graphed above?

Answer

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

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

We can now find the -intercept form of the line by substituting 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 :

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Question

Which of the following inequalities is graphed above?

Answer

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

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

We can now find the -intercept form of the line by substituting $m=\frac{1}{2}$ and the point into the slope-intercept equation and solving for :

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

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 :

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

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

___

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

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Question

Which of the following inequalities is graphed above?

Answer

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 , 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 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 :

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