Solving Inequalities Word Problems
Inequalities word problems are a crucial part of algebra, and they can be a bit tricky to solve. But don’t worry, with the right approach and some practice, you’ll become a pro in no time. In this post, we’ll walk you through some steps and examples to help you solve inequalities word problems with ease.
What are Inequalities Word Problems?
Inequalities word problems are algebraic expressions that involve unknown values or variables. They are used to represent real-world situations where the relationship between the variables is not equal, but rather greater than, less than, or equal to. These problems can be found in various fields, such as physics, economics, and engineering.
Types of Inequalities Word Problems
There are several types of inequalities word problems, including:
- Linear Inequalities: These involve a single variable and a constant. For example, 2x + 3 > 5.
- Non-Linear Inequalities: These involve more than one variable or a non-linear relationship. For example, x^2 + 3x - 4 > 0.
- Compound Inequalities: These involve multiple inequalities joined by “and” or “or.” For example, 2x + 3 > 5 and x - 2 < 3.
Steps to Solve Inequalities Word Problems
To solve inequalities word problems, follow these steps:
- Read the Problem Carefully: Read the problem statement carefully and identify the variables and constants involved.
- Translate the Problem: Translate the problem into an algebraic expression using the variables and constants identified in step 1.
- Simplify the Expression: Simplify the expression by combining like terms and canceling out any common factors.
- Solve the Inequality: Solve the inequality using the rules of algebra, such as adding, subtracting, multiplying, or dividing both sides by a non-zero value.
- Check the Solution: Check the solution by plugging it back into the original equation to ensure it satisfies the inequality.
Examples of Inequalities Word Problems
Here are some examples of inequalities word problems:
Example 1: Linear Inequality
Tom has been saving money for a new bike and has 120 in his savings account. He wants to buy a bike that costs 180. If he earns $15 per hour by mowing lawns, how many hours does he need to work to have enough money to buy the bike?
Let x be the number of hours Tom needs to work. The inequality can be written as:
15x + 120 ≥ 180
Simplifying the expression, we get:
15x ≥ 60
Dividing both sides by 15, we get:
x ≥ 4
Therefore, Tom needs to work at least 4 hours to have enough money to buy the bike.
Example 2: Non-Linear Inequality
A bakery sells a total of 250 loaves of bread per day. They sell a combination of whole wheat and white bread. If they sell at least 30 more whole wheat loaves than white bread loaves, and they sell 20 more whole wheat loaves than the day before, how many whole wheat loaves did they sell today?
Let x be the number of whole wheat loaves sold today. The inequality can be written as:
x ≥ 30 + (x - 20)
Simplifying the expression, we get:
x ≥ 50
Therefore, the bakery sold at least 50 whole wheat loaves today.
Example 3: Compound Inequality
A farmer has 100 acres of land to plant crops. He wants to plant at least 20 acres of wheat and at most 30 acres of corn. If he plants 10 acres of soybeans, how many acres of wheat and corn can he plant?
Let x be the number of acres of wheat and y be the number of acres of corn. The inequality can be written as:
x + y + 10 ≤ 100 x ≥ 20 y ≤ 30
Simplifying the expressions, we get:
x + y ≤ 90 x ≥ 20 y ≤ 30
Therefore, the farmer can plant at least 20 acres of wheat and at most 30 acres of corn.
Table of Inequalities
Here is a table summarizing the inequalities we’ve discussed:
| Type of Inequality | Example | Solution |
|---|---|---|
| Linear Inequality | 15x + 120 ≥ 180 | x ≥ 4 |
| Non-Linear Inequality | x ≥ 30 + (x - 20) | x ≥ 50 |
| Compound Inequality | x + y + 10 ≤ 100, x ≥ 20, y ≤ 30 | x + y ≤ 90, x ≥ 20, y ≤ 30 |
Notes
- When solving inequalities, make sure to check the solution by plugging it back into the original equation.
- When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality.
- When solving compound inequalities, make sure to satisfy all the conditions.
Inequalities Word Problems Worksheet
Here are some inequalities word problems for you to practice:
- A bookshelf has 5 shelves, and each shelf can hold at most 8 books. If the bookshelf is currently empty, how many books can be placed on it in total?
- A bakery sells a total of 300 cupcakes per day. They sell a combination of chocolate and vanilla cupcakes. If they sell at least 20 more chocolate cupcakes than vanilla cupcakes, how many chocolate cupcakes did they sell today?
- A farmer has 150 acres of land to plant crops. He wants to plant at least 30 acres of wheat and at most 40 acres of corn. If he plants 10 acres of soybeans, how many acres of wheat and corn can he plant?
Answers
- 40 books
- At least 20 chocolate cupcakes
- At least 30 acres of wheat and at most 40 acres of corn
Frequently Asked Questions
What is an inequality?
+An inequality is a statement that two values are not equal.
How do I solve an inequality?
+To solve an inequality, read the problem carefully, translate the problem into an algebraic expression, simplify the expression, and solve the inequality using the rules of algebra.
What is a compound inequality?
+A compound inequality is an inequality that involves multiple inequalities joined by "and" or "or."
In conclusion, inequalities word problems can be challenging, but with practice and the right approach, you can become proficient in solving them. Remember to read the problem carefully, translate the problem into an algebraic expression, simplify the expression, and solve the inequality using the rules of algebra. Happy solving!
Related Terms:
- Translating Inequalities Word problems Worksheet
