What are 'and/or' inequalities in math, better known as compound inequalities? Learn how to solve compound inequalities and how to graph compound inequalities. Updated: 02/07/2022 A compound inequality is made up of two inequalities connected by the word 'and' or the word 'or.' Compound inequalities are also called 'and/or,' 'and & or,' and 'and-or' inequalities. Compound inequalities describe the relationship of one value to two other values. In mathematics, inequalities are like equations with the difference that the answer is a range of numbers instead of just a specific number. Consider the following scenario: Peter is creating a rectangular vegetable garden in his backyard. The length of the garden is 10 feet. The perimeter of the garden must be at least 34 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden. Compound Inequalities
Compound Inequalities: OrThe term disjunction is also used to describe an 'Or' inequality. In other words, the range of values that satisfy either part of the inequality does not connect on the number line. Whenever 'Or' compound inequalities are solved, the goal is to find a true statement that meets either inequality or both inequality conditions. For example, x > 7 OR x < 3, when graphing these inequalities on a number line, it will leave a gap between the numbers 3 and 7. Therefore, 'Or' compound inequalities produce two separate sets of possible values. Compound Inequalities: AndAdditionally, an 'And' compound inequality is also known as a conjunction. These types of compound inequalities are written as a long inequality like 3 > x > 9. This is the same as 3>x and x > 9. In this and other compound inequalities, the answers are a range of possible values between 3 and 9. When solving 'And' inequalities, the range of values must satisfy both conditions. These inequalities show a range of values that overlap. What Are Inequalities?Sydney has an important math test coming up on Friday. She wants to study for her test for at least 30 minutes. Sydney's parents told her she needs to leave for swimming practice in 60 minutes. Therefore, she can study for no less than 30 minutes and no more than 60 minutes. This scenario can be written as a compound inequality. But what exactly does this mean? You can think of an inequality as an equation, except that the equals sign is replaced with a less than or greater than sign. We still need to solve the inequality just like you would an equation. The only difference is that instead of one answer that makes the equation true, like x = 3, there are many answers that make an inequality true, like x < 5. In this case, all numbers less than five would make the inequality true.
A compound inequality is just more than one inequality that we want to solve at the same time. We can either use the word 'and' or 'or' to indicate if we are looking at the solution to both inequalities (and), or if we are looking at the solution to either one of the inequalities (or). x < 7 and x > -3, which can also be written as -3 < x < 7, is a compound inequality because it is two inequalities connected by the word 'and'. This is also known as a conjunction. In this case, we are looking for the solution to both inequalities. In other words, this solution satisfies both inequalities. x > 7 or x < -3 is a compound inequality, also known as a disjunction, because it is two inequalities connected by the word 'or'. In this case, we are looking for a solution to either one of the equations. Let's check back in with Sydney. We know she needs to study for at least 30 minutes, but less than 60. If we set this up as a compound inequality, it looks like this: x > 30 and x < 60, also written as 30 < x < 60.
How to Solve Compound InequalitiesTo solve compound inequalities, follow the same procedure as solving equations. However, since compound inequalities are made up of two inequalities, separate them and solve each inequality independently. Once the inequalities are separated, isolate the variable by using the inverse operation just like what is done to solve equations. For example, to solve the compound inequality 14 > 2x > 4 follow these steps: 14 > 2x > 4 14 > 2x and 2x >4 separate the two inequalities 14/2> 2x/2 and 2x/2> 4/2 to isolate the variable, divide both sides by 2 7 > x and x > 2 there are now two sets of answers 7 > x > 2 the range of answers are between 2 and 7 Look at the first scenario again: Peter is creating a rectangular vegetable garden in his backyard. The length of the garden is 10 feet. The perimeter of the garden must be at least 34 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden. The following compound inequality can be used to represent the problem Inequalities 1
number line 1
To graph the results, use a number line and select the range of numbers between 7 and 14. Those numbers satisfy both inequalities. Graphing Compound Inequalities on a Number LineIn this section, the concept of how to graph compound inequalities on a number line will be explained. To accomplish this, pay attention to the signs. If the sign is less than or greater, then use an open circle and shade in or color everything to the right or to the left of the number. If the sign is less than or equal to or greater than or equal to, then shade the circle completely because it means the number is included in the answer. These rules can be extended from single to compound inequalities. Therefore, to graph the compound inequality -1 > x or x > 3 follow these steps: Put an open circle on the negative 1 Shade the area to the left of negative 1 Put an open circle on the positive 3 Shade the area to the right of positive 3 Number line 2
How to Solve a Compound InequalityExample 1Let's take a look at the inequality 2 + x < 5 and -1 < 2 + x, which can also be written as -1 < 2 + x < 5. This is a compound inequality because it uses the word 'and.' Now let's go ahead and solve it. 1) Solve each part of the inequality separately. 2 + x < 5 and -1 < 2 + x In the first equation, 2 + x < 5, we need to subtract 2 from each side to get the variable by itself. We then get x < 3. In the second equation, -1 < 2 + x, we again subtract 2 from both sides. This gives us -3 < x. Our solution, then, is x < 3 and -3 < x, or -3 < x < 3. 2) Graph on the number line.  Since this is a conjunction, the space between -3 and 3 is where the answer lies. In other words, any value between -3 and 3 satisfies this compound inequality. Remember Sydney? If we were to display her inequality on a number line, it would show that all numbers between 30 and 60 would be possible solutions. Meaning, she could study for 35, minutes, 42 minutes and so on. Example 2But what if we are solving a disjunction? Let's take a look at the following inequality: 7 > 2x + 5 or 7 < 5x - 3. This time the word 'or' is used instead of the word 'and'. How do we solve this? 1) Solve each inequality: For 7 > 2x + 5, we subtract 5 from each side to get 2 > 2x. Divide each side by 2 and we get 1 > x. For 7 < 5x - 3, we add 3 to each side and get 10 < 5x. Divide each side by 5 and we have 2 < x. 2) Graph on the number line. Since this is a disjunction, any value greater than 2 and less than 1 is where the answer lies. All real numbers satisfy this compound inequality. Important NotesThere are a few important points to keep in mind as you solve compound inequalities. What Are Inequalities?Sydney has an important math test coming up on Friday. She wants to study for her test for at least 30 minutes. Sydney's parents told her she needs to leave for swimming practice in 60 minutes. Therefore, she can study for no less than 30 minutes and no more than 60 minutes. This scenario can be written as a compound inequality. But what exactly does this mean? You can think of an inequality as an equation, except that the equals sign is replaced with a less than or greater than sign. We still need to solve the inequality just like you would an equation. The only difference is that instead of one answer that makes the equation true, like x = 3, there are many answers that make an inequality true, like x < 5. In this case, all numbers less than five would make the inequality true.
A compound inequality is just more than one inequality that we want to solve at the same time. We can either use the word 'and' or 'or' to indicate if we are looking at the solution to both inequalities (and), or if we are looking at the solution to either one of the inequalities (or). x < 7 and x > -3, which can also be written as -3 < x < 7, is a compound inequality because it is two inequalities connected by the word 'and'. This is also known as a conjunction. In this case, we are looking for the solution to both inequalities. In other words, this solution satisfies both inequalities. x > 7 or x < -3 is a compound inequality, also known as a disjunction, because it is two inequalities connected by the word 'or'. In this case, we are looking for a solution to either one of the equations. Let's check back in with Sydney. We know she needs to study for at least 30 minutes, but less than 60. If we set this up as a compound inequality, it looks like this: x > 30 and x < 60, also written as 30 < x < 60. How to Solve a Compound InequalityExample 1Let's take a look at the inequality 2 + x < 5 and -1 < 2 + x, which can also be written as -1 < 2 + x < 5. This is a compound inequality because it uses the word 'and.' Now let's go ahead and solve it. 1) Solve each part of the inequality separately. 2 + x < 5 and -1 < 2 + x In the first equation, 2 + x < 5, we need to subtract 2 from each side to get the variable by itself. We then get x < 3. In the second equation, -1 < 2 + x, we again subtract 2 from both sides. This gives us -3 < x. Our solution, then, is x < 3 and -3 < x, or -3 < x < 3. 2) Graph on the number line. Since this is a conjunction, the space between -3 and 3 is where the answer lies. In other words, any value between -3 and 3 satisfies this compound inequality. Remember Sydney? If we were to display her inequality on a number line, it would show that all numbers between 30 and 60 would be possible solutions. Meaning, she could study for 35, minutes, 42 minutes and so on. Example 2But what if we are solving a disjunction? Let's take a look at the following inequality: 7 > 2x + 5 or 7 < 5x - 3. This time the word 'or' is used instead of the word 'and'. How do we solve this? 1) Solve each inequality: For 7 > 2x + 5, we subtract 5 from each side to get 2 > 2x. Divide each side by 2 and we get 1 > x. For 7 < 5x - 3, we add 3 to each side and get 10 < 5x. Divide each side by 5 and we have 2 < x. 2) Graph on the number line. Since this is a disjunction, any value greater than 2 and less than 1 is where the answer lies. All real numbers satisfy this compound inequality. Important NotesThere are a few important points to keep in mind as you solve compound inequalities. How does one know if a compound inequality is 'And' or 'Or'?'Or' compound inequalities produce two separate sets of possible values. 'And' inequalities show a range of values that overlap on the number line. How are 'and/or' inequalities solved?To solve compound inequalities, follow the same procedure as solving equations. However, since compound inequalities are made up of two inequalities, separate them and solve each inequality independently. Once the inequalities are separated, isolate the variable by using the inverse operation just like what is done to solve equations. What does 'and/or' mean in inequalities?An 'And' inequality is also known as a conjunction. These types of compound inequalities are written as a long inequality such as 3 > x > 9. These inequalities show a range of values that overlap. The term disjunction is also used to describe an 'Or' inequality. In other words, the range of values that satisfy either part of the inequality does not connect on the number line. Whenever 'Or' compound inequalities are solved, the goal is to find a true statement that meets either inequality or both inequality conditions. Therefore, 'Or' compound inequalities produce two separate sets of possible values. Register to view this lessonAre you a student or a teacher? Unlock Your EducationSee for yourself why 30 million people use Study.comBecome a Study.com member and start learning now.Become a Member Already a member? Log In Back Resources created by teachers for teachersOver 30,000 video lessons & teaching resources‐all in one place. Video lessons Quizzes & Worksheets Classroom Integration Lesson Plans I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline. Back Create an account to start this course today Used by over 30 million students worldwide Create an account How do you find the compound inequality?To find the solution of an "and" compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs—where the graphs overlap. For the compound inequality x>−3 and x≤2, we graph each inequality. We then look for where the graphs “overlap”.
How are compound inequalities represented?A compound inequality contains at least two inequalities that are separated by either "and" or "or". The graph of a compound inequality with an "and" represents the intersection of the graph of the inequalities. A number is a solution to the compound inequality if the number is a solution to both inequalities.
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Write a compound inequality that is represented by the graph
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