How to graph a two-step inequality - Geometry

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Question

Points and lie on a circle. Which of the following could be the equation of that circle?

Answer

If we plug the points and into each equation, we find that these points work only in the equation $(x-3)^{2}+(y-3)^{2}=9$ . This circle has a radius of and is centered at .

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Question

Let D be the region on the (x,y) coordinate plane that contains the solutions to the following inequalities:

$x\leq k$ , where is a positive constant

$0\leq y\leq 12x$

Which of the following expressions, in terms of ___, is equivalent to the area of D?

Answer

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Question

Solve and graph the following inequality:

$2x - 7 \geq 6$

Answer

To solve the inequality, the first step is to add to both sides:

$2x - 7 +7 \geq 6+7$

$2x \geq 13$

The second step is to divide both sides by :

$\frac{2x}{2} \geq \frac{13}{2}$

$x \geq 6.5$

To graph the inequality, you draw a straight number line. Fill in the numbers from to infinity. Infinity can be designated by a ray. Be sure to fill in the number , since the equation indicated greater than OR equal to.

The graph should look like:

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Question

Which of the following lines is perpendicular to the line ?

Answer

The key here is to look for the line whose slope is the negative reciprocal of the original slope.

In this case, $\frac{-1}{3}$ is the negative reciprocal of .

Therefore, the equation of the line which is perpendicular to the original equation is:

$y=\frac{-x}{3}+2$

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Question

Which inequality does this graph represent?

Answer

The two lines represented are $y=\frac{1}{2}x+1$ and $y=-\frac{3}5{}x+3$ . The shaded region is below both lines but above

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Question

What is the area of the shaded region for the following inequality:

$y\leq -\frac{3}{5}x+3$ ;

$0\leq y \leq \frac{1}{2}x+1$

Answer

This inequality will produce the following graph:

The shaded area is a triangle with base 7 and height 2.

To find the area, plug these values into the area formula for a triangle, $\frac{1}{2}bh$ .

In this case, we are evaluating $\frac{1}{2}\cdot 7\cdot 2$ , which equals 7.

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Question

What is the area of the shaded region for this system of inequalities:

$0\leq y \leq \frac{1}{2}x+3$ ; $\frac{1}{2}x \leq y \leq 5$

Answer

Once graphed, the inequality will look like this:

To find the area, it is easiest to consider it as 2 congruent triangles with base 6 and height 3.

The total area will then be

$2\cdot \frac{1}{2}(6)(3)$ , or just $6\cdot 3 = 18$ .

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Question

Find the -intercept for the following:

Answer

$When\ finding\ the\ y-intercept,\ you\ must\ plug\ in\ x=0\ and\ solve\ for\ y.$

$When\ x=0\ the\ equation\ becomes: 3(0)-2y=11$ .

$The\ next\ step\ becomes: -2y=11$ .

$Dividing\ both\ sides\ by\ -2\ gives\ us\ the\ y-int\ which\ is\ y=-\frac{11}{2}$ .

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Question

Points and lie on a circle. Which of the following could be the equation of that circle?

Answer

If we plug the points and into each equation, we find that these points work only in the equation $(x-3)^{2}+(y-3)^{2}=9$ . This circle has a radius of and is centered at .

Compare your answer with the correct one above

Question

Let D be the region on the (x,y) coordinate plane that contains the solutions to the following inequalities:

$x\leq k$ , where is a positive constant

$0\leq y\leq 12x$

Which of the following expressions, in terms of ___, is equivalent to the area of D?

Answer

Compare your answer with the correct one above

Question

Solve and graph the following inequality:

$2x - 7 \geq 6$

Answer

To solve the inequality, the first step is to add to both sides:

$2x - 7 +7 \geq 6+7$

$2x \geq 13$

The second step is to divide both sides by :

$\frac{2x}{2} \geq \frac{13}{2}$

$x \geq 6.5$

To graph the inequality, you draw a straight number line. Fill in the numbers from to infinity. Infinity can be designated by a ray. Be sure to fill in the number , since the equation indicated greater than OR equal to.

The graph should look like:

Compare your answer with the correct one above

Question

Which of the following lines is perpendicular to the line ?

Answer

The key here is to look for the line whose slope is the negative reciprocal of the original slope.

In this case, $\frac{-1}{3}$ is the negative reciprocal of .

Therefore, the equation of the line which is perpendicular to the original equation is:

$y=\frac{-x}{3}+2$

Compare your answer with the correct one above

Question

Which inequality does this graph represent?

Answer

The two lines represented are $y=\frac{1}{2}x+1$ and $y=-\frac{3}5{}x+3$ . The shaded region is below both lines but above

Compare your answer with the correct one above

Question

What is the area of the shaded region for the following inequality:

$y\leq -\frac{3}{5}x+3$ ;

$0\leq y \leq \frac{1}{2}x+1$

Answer

This inequality will produce the following graph:

The shaded area is a triangle with base 7 and height 2.

To find the area, plug these values into the area formula for a triangle, $\frac{1}{2}bh$ .

In this case, we are evaluating $\frac{1}{2}\cdot 7\cdot 2$ , which equals 7.

Compare your answer with the correct one above

Question

What is the area of the shaded region for this system of inequalities:

$0\leq y \leq \frac{1}{2}x+3$ ; $\frac{1}{2}x \leq y \leq 5$

Answer

Once graphed, the inequality will look like this:

To find the area, it is easiest to consider it as 2 congruent triangles with base 6 and height 3.

The total area will then be

$2\cdot \frac{1}{2}(6)(3)$ , or just $6\cdot 3 = 18$ .

Compare your answer with the correct one above

Question

Find the -intercept for the following:

Answer

$When\ finding\ the\ y-intercept,\ you\ must\ plug\ in\ x=0\ and\ solve\ for\ y.$

$When\ x=0\ the\ equation\ becomes: 3(0)-2y=11$ .

$The\ next\ step\ becomes: -2y=11$ .

$Dividing\ both\ sides\ by\ -2\ gives\ us\ the\ y-int\ which\ is\ y=-\frac{11}{2}$ .

Compare your answer with the correct one above

Question

Let D be the region on the (x,y) coordinate plane that contains the solutions to the following inequalities:

$x\leq k$ , where is a positive constant

$0\leq y\leq 12x$

Which of the following expressions, in terms of ___, is equivalent to the area of D?

Answer

Compare your answer with the correct one above

Question

Solve and graph the following inequality:

$2x - 7 \geq 6$

Answer

To solve the inequality, the first step is to add to both sides:

$2x - 7 +7 \geq 6+7$

$2x \geq 13$

The second step is to divide both sides by :

$\frac{2x}{2} \geq \frac{13}{2}$

$x \geq 6.5$

To graph the inequality, you draw a straight number line. Fill in the numbers from to infinity. Infinity can be designated by a ray. Be sure to fill in the number , since the equation indicated greater than OR equal to.

The graph should look like:

Compare your answer with the correct one above

Question

Which of the following lines is perpendicular to the line ?

Answer

The key here is to look for the line whose slope is the negative reciprocal of the original slope.

In this case, $\frac{-1}{3}$ is the negative reciprocal of .

Therefore, the equation of the line which is perpendicular to the original equation is:

$y=\frac{-x}{3}+2$

Compare your answer with the correct one above

Question

Points and lie on a circle. Which of the following could be the equation of that circle?

Answer

If we plug the points and into each equation, we find that these points work only in the equation $(x-3)^{2}+(y-3)^{2}=9$ . This circle has a radius of and is centered at .

Compare your answer with the correct one above

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