How To Graph Compound Inequality

Compound Inequalities

Learning Objectives

· Solve compound inequalities in the form of or and express the solution graphically.

· Solve compound inequalities in the form of and and express the solution graphically.

· Solve chemical compound inequalities in the class a < x < b.

· Place cases with no solution.

Introduction

Many times, solutions lie between two quantities, rather than continuing endlessly in one management. For example systolic (top number) blood pressure that is between 120 and 139 mm Hg is called borderline high claret pressure. This tin be described using a compound inequality, b < 139 and b > 120. Other compound inequalities are joined by the word "or".

When two inequalities are joined by the word and, the solution of the compound inequality occurs when both inequalities are true at the aforementioned fourth dimension. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word or, the solution of the compound inequality occurs when either of the inequalities is true. The solution is the combination, or union, of the two private solutions.

Solving and Graphing Compound Inequalities in the Class of "or"

Let's take a closer wait at a chemical compound inequality that uses or to combine two inequalities. For example, 10 > six or ten < 2. The solution to this compound inequality is all the values of x in which x is either greater than six or x is less than two. You can show this graphically past putting the graphs of each inequality together on the same number line.

The graph has an open circle on 6 and a blue pointer to the right and another open up circle at 2 and a cherry-red arrow to the left. In fact, the but parts that are not a solution to this chemical compound inequality are the points 2 and half-dozen and all the points in between these values on the number line. Everything else on the graph is a solution to this compound inequality.

Let's look at another example of an or compound inequality, x > 3 or x ≤ 4. The graph of x > 3 has an open circumvolve on 3 and a blue pointer drawn to the right to contain all the numbers greater than 3.

The graph of x ≤ 4 has a airtight circumvolve at 4 and a red arrow to the left to contain all the numbers less than 4.

What do you notice most the graph that combines these two inequalities?

Since this compound inequality is an or statement, it includes all of the numbers in each of the solutions, which in this case is all the numbers on the number line. (The region of the line greater than three and less than or equal to four is shown in purple considering it lies on both of the original graphs.) The solution to the chemical compound inequality 10 > 3 or x ≤ iv is the set of all real numbers!

Yous may demand to solve i or more of the inequalities before determining the solution to the compound inequality, as in the example below.

Example
Problem	Solve for ten. threex – 1 < 8 or x – 5 > 0
		Solve each inequality by isolating the variable. Write both inequality solutions equally a compound using or.
Reply

The solution to this chemical compound inequality can be shown graphically.

Call up to utilize the properties of inequality when you are solving chemical compound inequalities. The next example involves dividing past a negative to isolate a variable.

Example
Problem	Solve for y. twoy + 7 < thirteen or −3y – two 10
			Solve each inequality separately. The inequality sign is reversed with division by a negative number. Since y could be less than iii or greater than or equal to −4, y could exist any number.
Answer		The solution is all real numbers.

This number line shows the solution set of y < 3 or y ≥ four.

Example
Problem	Solve for z . 5z – 3 > −18 or −2z – 1 > 15
		Solve each inequality separately. Combine the solutions.
Answer

This number line shows the solution set of z > −3 or z < −viii.

Solve for h.

h + iii > 12 or 3 – 2h > 9

A) h < 3 or h > − 3

B) h > 9 or h > − 3

C) h > − 9 or h < 3

D) h > nine or h < − 3

Bear witness/Hibernate Reply

A) h < 3 or h > − 3

Incorrect. To solve the inequality h + 3 > 12, subtract 3 from both sides to get h > 9. When you dissever both sides of an inequality by a negative number, reverse the inequality sign to get h < − iii for the solution to the second inequality. The right reply is h > ix or h < − 3.

B) h > 9 or h > − 3

Incorrect. To solve the inequality three – 2h > 9, subtract 3 from both sides and then divide by − two. When you divide both sides of an inequality past a negative number, reverse the inequality sign to get h < − 3. The right answer is h > 9 or h < − 3.

C) h > − ix or h < 3

Incorrect. Bank check a few values for h that are greater than − 9 just less than iii, and come across if they brand the inequality true. For example, if you lot substitute h = 2 into each inequality, you get false statements: two + three > 9; 3 – ii(two) > 9. The correct respond is h > 9 or h < − 3.

D) h > nine or h < − three

Right. Solving each inequality for h, you find that h > 9 or h < − 3.

Solving and Graphing Compound Inequalities in the Form of "and"

The solution of a compound inequality that consists of two inequalities joined with the give-and-take and is the intersection of the solutions of each inequality. In other words, both statements must be truthful at the same time. The solution to an and compound inequality are all the solutions that the two inequalities have in common. Graphically, y'all can call back nearly it as where the ii graphs overlap.

Think most the instance of the chemical compound inequality: x < five and x ≥ − 1. The graph of each individual inequality is shown in color.

Since the word and joins the two inequalities, the solution is the overlap of the 2 solutions. This is where both of these statements are true at the same fourth dimension.

The solution to this compound inequality is shown below.

Notice that in this case, you lot tin can rewrite x ≥ − 1 and x < v as − 1 ≤ ten < 5 since the solution is betwixt − one and 5, including − 1. You read − 1 ≤ x < 5 as "x is greater than or equal to − ane and less than five." You can rewrite an and statement this manner only if the answer is between two numbers.

Permit'due south await at two more examples.

Example
Problem	Solve for x.
		Solve each inequality for 10. Determine the intersection of the solutions.
The number line below shows the graphs of the 2 inequalities in the problem. The solution to the compound inequality is x ≥ four, as this is where the 2 graphs overlap. And the solution can exist represented as:
Respond

Case
Trouble	Solve for x.
		Solve each inequality separately. Notice the overlap between the solutions.
The ii inequalities can be represented graphically equally: And the solution can be represented equally:
Respond

Rather than splitting a compound inequality in the form of a < x < b into two inequalities x < b and x > a, yous can more quickly solve the inequality past applying the properties of inequality to all three segments of the compound inequality. Two examples are provided below.

Example
Problem	Solve for x.
		Isolate the variable by subtracting 3 from all 3 parts of the inequality, and then dividing each part past 2.
Reply

Case
Problem	Solve for 10.
		Isolate the variable by subtracting 7 from all 3 parts of the inequality, and and so dividing each part by 2.
Answer

To solve inequalities like a < 10 < b, apply the addition and multiplication properties of inequality to solve the inequality for x. Whatever operation you lot perform on the heart portion of the inequality, you must too perform to each of the outside sections as well. Pay particular attention to division or multiplication past a negative.

Which of the post-obit compound inequalities represents the graph on the number line below?

A) − 8 ≥ x > − one

B) − 8 ≤ ten < − 1

C) − eight ≤ ten > − 1

D) − 8 ≥ x < − 1

Show/Hide Answer

A) − 8 ≥ ten > − 1

Wrong. This compound inequality reads, "ten is less than or equal to − 8 and greater than − one." The shaded part of the graph includes values that are greater than or equal to − 8 and less than − 1. The correct answer is − 8 ≤ x < − 1.

B) − 8 ≤ x < − ane

Correct. The selected region on the number line lies betwixt − eight and − 1and includes -8, then x must be greater than or equal to − 8 and less than − i.

C) − viii ≤ x > − ane

Incorrect. This compound inequality reads, "x is greater than or equal to − 8 and greater than − i." The values that are shaded are less − one, not greater. The correct answer is − eight ≤ x < − 1.

D) − 8 ≥ ten < − 1

Incorrect. This compound inequality reads, "x is less than or equal to − 8 and less than − 1." The graph does not include values that are less than or equal to − 8. It includes values that are greater than or equal to − 8 and less than − ane. The correct answer is − 8 ≤ ten < − ane.

Special Cases of Compound Inequalities

The solution to a compound inequality with and is always the overlap between the solution to each inequality. In that location are three possible outcomes for compound inequalities joined past the discussion and:

1. The solution could be all the values between 2 endpoints.

2. The solution could begin at a point on the number line and extend in i direction.

3. In cases where there is no overlap between the 2 inequalities, there is no solution to the compound inequality.

An example is shown beneath.

Instance
Problem	Solve for x. *ten* + 2 > five and 10 + 4 < v
		Solve each inequality separately. Discover the overlap betwixt the solutions.

Answer There is no overlap between , and then there is no solution.

Summary

A chemical compound inequality is a statement of two inequality statements linked together either by the word or or by the word and. Sometimes, an and compound inequality is shown symbolically, like a < x < b, and does not even demand the word and. Considering compound inequalities represent either a spousal relationship or intersection of the private inequalities, graphing them on a number line can exist a helpful way to see or check a solution. Compound inequalities can be manipulated and solved much the same way any inequality is solved, paying attention to the backdrop of inequalities and the rules for solving them.