How to Solve an Inequality: A Comprehensive Guide with Examples

Introduction

An inequality is a mathematical statement that compares two or more quantities, expressing that they are not equal. Inequalities are widely used in mathematics, science, engineering, and economics. Knowing how to solve inequalities is essential if you want to understand these fields and excel in academic settings. In this article, we’ll provide you with a comprehensive guide to solving inequalities, from the basics to applications in calculus. We’ll also give you tips on how to avoid common mistakes and check your answers.

A Step-by-Step Guide to Solving Inequalities

The first step in solving an inequality is to identify the variable (or variables) and the inequality sign(s). After that, you’ll follow these steps:

1. Simplify both sides of the inequality as much as possible.
2. Isolate the variable on one side of the inequality. Move constants to the other side using inverse operations (i.e., add or subtract on both sides).
3. When multiplying or dividing both sides by a negative number, flip the inequality sign.
4. If the variable is surrounded by absolute value signs, break the inequality into two separate inequalities and solve each one.

Let’s see some examples:

One-variable inequality:

x + 3 > 7

1. Subtract 3 from both sides: x > 4

Two-variable inequality:

4x – 3y ≤ 2

1. Add 3y to both sides: 4x ≤ 2 + 3y
2. Divide both sides by 4: x ≤ 1/2 + 3/4y

Always remember to check your answer by plugging it back into the original inequality.

Graphing Inequalities

If you’re a visual learner, graphing inequalities on a number line can be a helpful tool for solving them. Here’s how to do it:

1. Draw a number line and label it appropriately.
2. Identify the variable and the inequality sign(s).
3. Shade the region of the number line that satisfies the inequality:
• For < or > signs, use open circles to show the endpoint is not included in the solution. Draw an arrow in the direction of the larger values.
• For ≤ or ≥ signs, use closed circles to show the endpoint is included in the solution. Draw an arrow in both directions.

Let’s see an example:

x + 2 < 5

Graph:

As we can see, x satisfies the inequality for all numbers less than 3.

Solving Compound Inequalities

Compound inequalities are inequalities that involve multiple variables or multiple inequality signs. Here’s how to solve them:

1. Solve each inequality separately.
2. Combine the solutions using the appropriate sign (& for “and” or | for “or”).

Let’s see some examples:

x < 2 or x > 5

Solution: x < 2 | x > 5

3 ≤ x + 2 < 7

Solution: 1 ≤ x < 5

Remember to check your answer by plugging it back into the original inequality.

Applications of Inequalities

Inequalities are widely used in real-life scenarios, such as in economics, finance, engineering, physics, and biology. Here are some examples:

• Determining the maximum profit a business can make based on the cost of production and the selling price of their product.
• Determining the minimum interest rate a bond should offer to be competitive with other investment options.
• Determining the maximum weight a bridge can support based on its capacity and the weight of the vehicles crossing it.
• Determining the minimum dosage of a medication needed to be effective based on a patient’s weight and age.

Solving inequalities is crucial in these fields to optimize cost, maximize profit, and ensure safety and efficiency.

Inequality Word Problems

Inequality word problems can be tricky, but they are essential in real-life scenarios. Here’s how to approach them:

1. Read the problem carefully and identify the variables involved.
2. Translate the problem into an inequality, using the inequality sign provided.
3. Solve the inequality using the methods we’ve covered so far.
4. Check your answer and make sure it makes sense in the context of the problem.

Let’s see some examples:

“You need at least $3000 in savings to afford a trip to Japan. You currently have $2000 in savings and plan to save $100 per month. In how many months will you be able to afford the trip?”

1. Variables: savings (S), months (m)
2. S ≥ 3000
3. 2000 + 100m ≥ 3000
4. m ≥ 10

It will take at least 10 months to afford the trip.

Common Mistakes to Avoid When Solving Inequalities

Here are some common errors people make when solving inequalities:

• Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
• Switching the direction of the inequality sign when adding or subtracting to both sides.
• Not checking whether an endpoint should be included or excluded from the solution.
• Not simplifying both sides of the inequality before solving it.

To avoid these mistakes, make sure you take your time and double-check your work.

Inequalities in Calculus

Inequalities play an essential role in calculus, particularly in optimization problems. Here’s an example of how to use inequalities to solve a calculus problem:

“Find the maximum area of a rectangle with a perimeter of 24 units.”

1. Variables: length (l), width (w)
2. Perimeter: 2l + 2w = 24
3. Length: l = 12 – w
4. Area: A = lw = (12 – w)w = 12w – w²
5. Take the derivative: A’ = 12 – 2w
6. Set equal to 0: 12 – 2w = 0, w = 6
7. Check the second derivative: A” = -2 < 0, so w = 6 is a maximum.
8. Substitute back into the equation for length: l = 6
9. The maximum area is 36 square units.

Inequalities are crucial in calculus to find the limit, determine the convergence of a series, and optimize functions.

Conclusion

Inequalities are a fundamental concept in mathematics and have numerous applications in real-life scenarios. In this article, we’ve covered the basics of solving inequalities, graphing them, solving compound inequalities, and tackling word problems. We’ve also given you tips on how to check your answer and avoid common mistakes. Finally, we’ve shown you how inequalities are used in calculus to optimize functions. Knowing how to solve inequalities is an essential skill for success in academic and professional settings.