SAT Mathematics : Graphing Inequalities

Study concepts, example questions & explanations for SAT Mathematics

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Example Questions

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Example Question #1 : Graphing Inequalities

For the following system of equations, what shape does the corresponding graph make:

Possible Answers:

Rectangle

It does not form a shape.

Square

Triangle

Correct answer:

Triangle

Explanation:

When we graph the three inequalities as equations, we get the below graph. Testing a point within the triangle does in fact fulfill all three conditions, so the shape formed by the inequalities is a triangle. If a tested point within the triangle did not fulfill one or more conditions, then we would have chosen “it does not form a shape.”

Screen shot 2020 09 29 at 10.05.26 am

Example Question #1 : Graphing Inequalities

Which of the following graphs correctly describes the system of inequalities:

 

Possible Answers:

Screen shot 2020 09 29 at 10.12.10 am

Screen shot 2020 09 29 at 10.11.35 am

Screen shot 2020 09 29 at 10.12.36 am

Screen shot 2020 09 29 at 10.10.52 am

Correct answer:

Screen shot 2020 09 29 at 10.11.35 am

Explanation:

After plotting the lines, shade the regions corresponding to each individual inequality. The area of intersection (in purple) is the solution to the system of inequalities. Another way to check your answer is to pick a point in all four regions delineated by the two equations and test those four coordinate points in the system of inequalities.

Example Question #1 : Graphing Inequalities

For the following system of equations, what shape does the corresponding graph make:

Possible Answers:

Square

It does not form a shape

Rectangle

Triangle

Correct answer:

Triangle

Explanation:

When we graph the three inequalities as equations, we get the below graph. Testing a point within the triangle does in fact fulfill all three conditions, so the shape formed by the inequalities is a triangle. If a tested point within the triangle did not fulfill one or more conditions, then we would have chosen “it does not form a shape.”

Screen shot 2020 09 29 at 11.41.58 am

Example Question #2 : Graphing Inequalities

Which of the following graphs correctly describes the system of inequalities:

 

Possible Answers:

Screen shot 2020 09 29 at 11.43.41 am

Screen shot 2020 09 29 at 11.43.45 am

Screen shot 2020 09 29 at 11.43.20 am

Screen shot 2020 09 29 at 11.43.50 am

Correct answer:

Screen shot 2020 09 29 at 11.43.41 am

Explanation:

After plotting the lines, shade the regions corresponding to each individual inequality. The area of intersection (in purple) is the solution to the system of inequalities. Another way to check your answer is to pick a point in all four regions delineated by the two equations and test those four coordinate points in the system of inequalities.

Example Question #1 : Graphing Inequalities

Which of the following ordered pairs (x,y) is a solution to the system of inequalities:

 

Possible Answers:

Correct answer:

Explanation:

This is the only coordinate point that fulfills the system of inequalities.

Example Question #6 : Graphing Inequalities

Screen shot 2020 09 29 at 11.46.39 am

If the system of inequalities  and  is graphed on the xy- plane above, which quadrant contains no solutions to the system?

Possible Answers:

III

II

I and II

I

Correct answer:

I and II

Explanation:

The graph of the system of inequalities is shown below. The solution (in purple) does not go into region I or II.

Screen shot 2020 09 29 at 11.48.13 am

Example Question #1 : Graphing Inequalities

Screen shot 2020 09 29 at 12.38.38 pm

If the system of inequalities  and  is graphed on the xy- plane above, which quadrant contains no solutions to the system?

Possible Answers:

There are solutions in all 4 quadrants.

III

II

IV

Correct answer:

There are solutions in all 4 quadrants.

Explanation:

The graph of the system of inequalities is shown below. The solution (in purple) is present in all four regions.

Screen shot 2020 09 29 at 12.40.48 pm

Example Question #8 : Graphing Inequalities

Screen shot 2020 09 29 at 12.41.34 pm

If the system of inequalities  and  is graphed on the xy- plane above, which quadrant contains no solutions to the system?

Possible Answers:

IV

There are solutions in all 4 quadrants.

III

II

Correct answer:

There are solutions in all 4 quadrants.

Explanation:

The graph of the system of inequalities is shown below. The solution (in purple) is present in all four regions.

Screen shot 2020 09 29 at 12.43.48 pm

Example Question #2 : Graphing Inequalities

Screen shot 2020 09 29 at 12.44.36 pm

The graph shown above is best represented by which inequality?

Possible Answers:

Correct answer:

Explanation:

The x-intercept is (4,0) so when  equals 0, , and the y-intercept is (0,3) so when  equals 0, . These points only work for a line corresponding to . To determine which direction the inequality points, you can plug in any point. If we use (0,0), , so this is the direction the inequality must face.

Example Question #4 : Graphing Inequalities

Which of the following are solutions to the system of equations  and ?

I) 

II) 

III) 

Possible Answers:

II

I

I, II, and III

I and II

Correct answer:

I

Explanation:

With this question, it is important to recognize the system of inequalities utilizes the “greater than” sign, not the “greater than or equal to” sign.  does not fulfill appears to fulfill both conditions but . 18 is not greater than 8, leaving only  as a possible solution.

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