# Solving Compound Inequalities Worksheet, Definition, and Examples

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### Key Points about Solving Compound Inequalities

- Compound inequalities involve two or more inequalities connected by “and” or “or,” essential in algebra and calculus.
- To solve compound inequalities, follow a step-by-step process: separate and solve each inequality, determine “and” or “or” for the solution, and visualize it on a number line.
- Common mistakes to avoid when solving compound inequalities include flipping inequality symbols, correctly combining solution sets, and checking solutions against the original compound inequality.

## What are Compound Inequalities?

Compound inequalities are a type of mathematical equation that involves two or more inequalities connected by the words “and” or “or.” Solving compound inequalities can be a tricky process, but it’s an essential skill for anyone studying algebra or calculus. In this article, we’ll explore how to solve compound inequalities, including examples and step-by-step instructions.

To solve compound inequalities, you’ll need to follow a few key steps. First, you’ll need to separate the inequalities and solve each one individually. Then, you’ll need to determine whether the answer should be “and” (union of the inequalities) or “or” (intersection of the inequalities). Finally, you’ll need to graph the solution on a number line to visualize the answer.

Whether you’re a student struggling with compound inequalities or a professional looking to brush up on your math skills, this article will provide you with the knowledge you need to solve these equations with confidence.

**Common Core Standard:**

**How to Solve Compound Inequalities**

**Step by Step Process**

Solving compound inequalities involves finding all values of the variable that make the compound inequality a true statement. There are two types of compound inequalities: those that involve the word “and” and those that involve the word “or.” The “and” compound inequality is true only if both inequalities are true, while the “or” compound inequality is true if at least one of the inequalities is true.

The following step-by-step process can be used to solve compound inequalities:

- Rewrite the compound inequality as two separate inequalities, connected by either “and” or “or.”
- Solve each inequality separately as if it were a simple inequality.
- Write the solution set for each inequality.
- Combine the solution sets using the appropriate conjunction, “and” or “or,” to obtain the solution set for the compound inequality.

For example, consider the compound inequality: 2x – 3 < 5 or 3x + 2 > 8

- Rewrite the compound inequality as two separate inequalities: 2x – 3 < 5 and 3x + 2 > 8
- Solve each inequality separately: 2x < 8 and 3x > 6
- Write the solution set for each inequality: x < 4 and x > 2
- Combine the solution sets using “or” since the original inequality used “or”: 2 < x < 4

**Common Mistakes**

There are some common mistakes that students make when solving compound inequalities. One common mistake is forgetting to switch the direction of the inequality when multiplying or dividing by a negative number. For example, if you multiply both sides of an inequality by -2, you must flip the direction of the inequality.

Another common mistake is forgetting to combine the solution sets using the appropriate conjunction. If the original inequality used “and,” the solution set must include only the values that satisfy both inequalities. If the original inequality used “or,” the solution set must include all values that satisfy at least one of the inequalities.

Finally, students often forget to check their solutions by plugging them back into the original inequality. This step is important to ensure that the solution set is accurate.

By following the step-by-step process and avoiding these common mistakes, students can successfully solve compound inequalities.

**Compound Inequalities Interval Notation**

Compound inequalities are inequalities that contain two or more inequalities joined by the words “and” or “or.” The solution to a compound inequality is the intersection or union of the solutions to the individual inequalities that make up the compound inequality.

**Types of Compound Inequalities**

There are two types of compound inequalities: “and” inequalities and “or” inequalities.

**And Inequalities**

An “and” inequality is a compound inequality that includes the word “and” between the two inequalities. The solution to an “and” inequality is the intersection of the solutions to the individual inequalities. In interval notation, the solution is expressed as an interval that satisfies both inequalities.

For example, consider the compound inequality:

2x – 1 < 5 and x + 3 > 1

To solve this inequality, we first solve each inequality separately:

2x < 6

x > -2

The solution to the compound inequality is the intersection of the solutions to the individual inequalities, which is:

x > -2 and x < 4

In interval notation, the solution is expressed as:

(-2, 4)

**Or Inequalities**

An “or” inequality is a compound inequality that includes the word “or” between the two inequalities. The solution to an “or” inequality is the union of the solutions to the individual inequalities. In interval notation, the solution is expressed as the union of two or more intervals that satisfy the individual inequalities.

For example, consider the compound inequality:

2x – 1 < 5 or x + 3 > 1

To solve this inequality, we first solve each inequality separately:

2x < 6

x > -2

The solution to the compound inequality is the union of the solutions to the individual inequalities, which is:

x < 3 or x > -2

In interval notation, the solution is expressed as:

(-∞, -2) U (3, ∞)

**Compound Inequalities Definition**

Compound inequalities are mathematical expressions that combine two or more simple inequalities. They can be written using the words “and” or “or” to connect the inequalities.

**And Compound Inequalities**

An “and” compound inequality is an expression that requires both inequalities to be true. It can be written in the form of “a < x < b” or “x is between a and b.” This means that x must be greater than a and less than b.

For example, if a = 2 and b = 6, then the “and” compound inequality would be “2 < x < 6.” This means that x must be greater than 2 and less than 6.

**Or Compound Inequalities**

An “or” compound inequality is an expression that requires only one of the inequalities to be true. It can be written in the form of “x < a or x > b” or “x is less than a or greater than b.” This means that x can either be less than a or greater than b, but not both.

For example, if a = 2 and b = 6, then the “or” compound inequality would be “x < 2 or x > 6.” This means that x can be less than 2 or greater than 6, but not between 2 and 6.

Compound inequalities can be graphed on a number line to help visualize the solution set. The solution set is the set of all values that satisfy the compound inequality.

**Compound Inequalities Examples**

Compound inequalities involve two or more inequalities joined by the words “and” or “or.” To solve a compound inequality, one must find all values of the variable that satisfy both inequalities simultaneously. Below are some examples of compound inequalities:

**Example 1:**

Solve the compound inequality 2x – 3 < 7 or 3x + 5 > 8.

**Solution:**

We can solve this compound inequality by solving each inequality separately and then combining the solutions.

For the first inequality, we have:

2x – 3 < 7

Adding 3 to both sides, we get:

2x < 10

Dividing both sides by 2, we get:

x < 5

For the second inequality, we have:

3x + 5 > 8

Subtracting 5 from both sides, we get:

3x > 3

Dividing both sides by 3, we get:

x > 1

Therefore, the solution to the compound inequality is:

1 < x < 5

**Example 2:**

Solve the compound inequality 2x – 3 > 7 and 3x + 5 < 8.

**Solution:**

We can solve this compound inequality by solving each inequality separately and then finding the intersection of the solutions.

For the first inequality, we have:

2x – 3 > 7

Adding 3 to both sides, we get:

2x > 10

Dividing both sides by 2, we get:

x > 5

For the second inequality, we have:

3x + 5 < 8

Subtracting 5 from both sides, we get:

3x < 3

Dividing both sides by 3, we get:

x < 1

Therefore, the solution to the compound inequality is:

x < 1 or x > 5

**Example 3:**

Solve the compound inequality -2 ≤ x + 3 < 5.

**Solution:**

We can solve this compound inequality by solving each inequality separately and then finding the intersection of the solutions.

For the first inequality, we have:

-2 ≤ x + 3

Subtracting 3 from both sides, we get:

-5 ≤ x

For the second inequality, we have:

x + 3 < 5

Subtracting 3 from both sides, we get:

x < 2

Therefore, the solution to the compound inequality is:

-5 ≤ x < 2

**Graphing Compound Inequalities**

Graphing compound inequalities involves plotting the solutions of two or more inequalities on the same number line. This helps to visualize the solution set of the compound inequality.

To graph a compound inequality, you first need to graph the solutions of each inequality separately. This is done by plotting the critical values on the number line and shading the appropriate region. Then, you combine the shaded regions to obtain the solution set of the compound inequality.

For instance, consider the compound inequality x < -2 or x > 3. To graph this inequality, you start by graphing the two simple inequalities x < -2 and x > 3 separately. The solution set of x < -2 is all values less than -2, which can be represented by shading the region to the left of -2 on the number line. Similarly, the solution set of x > 3 is all values greater than 3, which can be represented by shading the region to the right of 3 on the number line.

Next, you combine the shaded regions of the two simple inequalities to obtain the solution set of the compound inequality x < -2 or x > 3. The shaded regions of the two simple inequalities do not overlap, so the solution set of the compound inequality is all values less than -2 or greater than 3. This can be represented by shading the regions to the left of -2 and to the right of 3 on the number line.

It is important to note that when the compound inequality is connected by “and”, the solution set is the intersection of the solution sets of the simple inequalities. When the compound inequality is connected by “or”, the solution set is the union of the solution sets of the simple inequalities.

In summary, graphing compound inequalities involves plotting the solutions of two or more simple inequalities on the same number line and combining the shaded regions to obtain the solution set of the compound inequality.

**Compound Inequalities Word Problems**

Compound inequalities are used to represent a range of values that satisfy two or more inequalities. They are commonly used in word problems that involve ranges of values. In this section, we will look at some examples of compound inequality word problems and how to solve them.

**Example 1**

A company requires that all employees work between 25 and 40 hours per week. Write a compound inequality to represent this requirement.

To solve this problem, we need to represent the range of hours that an employee can work. We can do this by using the inequality symbols for “greater than or equal to” and “less than or equal to”. Therefore, the compound inequality is:

25 ≤ x ≤ 40

where x represents the number of hours worked per week.

**Example 2**

A store offers a discount on purchases of $100 or more and also offers a discount on purchases made with a store credit card. If a customer spends $100 or more and pays with a store credit card, they receive both discounts. Write a compound inequality to represent this situation.

To solve this problem, we need to represent the range of values that satisfy both conditions. We can use the inequality symbols for “greater than or equal to” and “less than or equal to” to represent the purchase amount and the credit card discount. Therefore, the compound inequality is:

p ≥ 100 and d ≥ 0.1p

where p represents the purchase amount and d represents the discount percentage.

**Example 3**

A school requires that all students maintain a grade point average (GPA) of at least 2.0 to remain in good standing. A student’s GPA is calculated by taking the average of their grades in all classes. If a student receives a grade of “A” in one class and a grade of “C” in another class, what is the minimum grade they need in their third class to maintain a GPA of at least 2.0?

To solve this problem, we need to represent the range of values that satisfy the requirement for a minimum GPA of 2.0. We can use the inequality symbol for “greater than or equal to” to represent the minimum GPA. Therefore, the compound inequality is:

(4 + 2 + x)/3 ≥ 2.0

where x represents the minimum grade needed in the third class. We can simplify this inequality by multiplying both sides by 3 to get:

6 + x ≥ 6

which simplifies to:

x ≥ 0

Therefore, the minimum grade the student needs in their third class is a “C” or higher.

Compound inequality word problems can be challenging, but by breaking them down into smaller parts and using the appropriate inequality symbols, they can be easily solved.

**Solving Compound Inequalities FAQ**

**How do you determine if a compound inequality is ‘and’ or ‘or’?**

To determine if a compound inequality is ‘and’ or ‘or,’ you need to look at the connecting word between the two inequalities. If the word is ‘and,’ then the solution must satisfy both inequalities. If the word is ‘or,’ then the solution must satisfy either one of the inequalities.

**What is an example of a compound inequality?**

An example of a compound inequality is “3 < x < 7.” This compound inequality can be read as “x is greater than 3 AND less than 7.” Another example is “x ≤ -2 OR x > 5,” which can be read as “x is less than or equal to -2 OR x is greater than 5.”

**How do you write a compound inequality?**

To write a compound inequality, you need to connect two simple inequalities with either ‘and’ or ‘or.’ For example, the compound inequality “3 < x < 7” can be written as “3 < x AND x < 7.” The compound inequality “x ≤ -2 OR x > 5” can be written as “x ≤ -2 OR x > 5.”

**What is the process for solving compound inequalities?**

The process for solving compound inequalities involves solving each simple inequality separately and then combining the solutions. If the compound inequality is connected with ‘and,’ then the solution must satisfy both inequalities. If the compound inequality is connected with ‘or,’ then the solution must satisfy either one of the inequalities.

**Where can I find extra practice problems for solving compound inequalities?**

Extra practice problems for solving compound inequalities can be found in algebra textbooks, online math forums, and educational websites such as Khan Academy and Mathway.

**What are some common mistakes to avoid when solving compound inequalities?**

Some common mistakes to avoid when solving compound inequalities include forgetting to distribute negative signs, forgetting to flip the inequality symbol when multiplying or dividing by a negative number, and not checking the solution in the original compound inequality.

**What are the signs of a compound inequality?**

The signs of a compound inequality are the inequality symbols that connect the two simple inequalities. The sign can be ‘and’ or ‘or.’

**How do you solve a compound inequality and statement?**

To solve a compound inequality and statement, you need to find the intersection of the solutions of the two simple inequalities. This means that the solution must satisfy both inequalities.

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