Bill Has Already Biked 4 Kilometers. He Will Bike More Than 5 Additional Kilometers. Let's Look At The Possible Total Numbers Of Kilometers Bill Will Bike.(a) Fill In The Blanks To Write An Inequality That Can Be Used To Find { X $}$, The
Introduction
In this article, we will explore the concept of inequalities and how they can be used to solve real-world problems. We will use the scenario of Bill's bike ride to illustrate the process of writing and solving inequalities.
The Problem
Bill has already biked 4 kilometers. He will bike more than 5 additional kilometers. Let's look at the possible total numbers of kilometers Bill will bike.
Step 1: Writing the Inequality
To find the total number of kilometers Bill will bike, we need to write an inequality that represents the situation. Let's use the variable x to represent the total number of kilometers Bill will bike.
Since Bill has already biked 4 kilometers, we can start by writing an inequality that represents the minimum number of kilometers he will bike:
x ≥ 4
This inequality states that the total number of kilometers Bill will bike (x) is greater than or equal to 4.
Step 2: Adding the Additional Kilometers
Bill will bike more than 5 additional kilometers. This means that the total number of kilometers he will bike (x) is greater than 4 + 5.
We can write this as an inequality:
x ≥ 4 + 5
Simplifying the inequality, we get:
x ≥ 9
This inequality states that the total number of kilometers Bill will bike (x) is greater than or equal to 9.
Step 3: Combining the Inequalities
We have two inequalities that represent the situation:
x ≥ 4 x ≥ 9
Since both inequalities must be true, we can combine them by using the "and" operator:
x ≥ 4 and x ≥ 9
This can be written as a single inequality:
x ≥ 9
Conclusion
In this article, we used the scenario of Bill's bike ride to illustrate the process of writing and solving inequalities. We started by writing an inequality that represented the minimum number of kilometers Bill would bike, and then added the additional kilometers to get the final inequality.
Solving the Inequality
Now that we have the inequality x ≥ 9, we can solve for x. Since the inequality is greater than or equal to 9, we can write:
x = 9 or x > 9
This means that the total number of kilometers Bill will bike (x) is either 9 or greater than 9.
Graphing the Inequality
We can graph the inequality x ≥ 9 by drawing a number line and marking the point 9. Since the inequality is greater than or equal to 9, we can shade the region to the right of 9.
Real-World Applications
Inequalities have many real-world applications, such as:
- Finance: Inequalities can be used to model financial situations, such as investments and loans.
- Science: Inequalities can be used to model scientific situations, such as population growth and chemical reactions.
- Engineering: Inequalities can be used to model engineering situations, such as stress and strain on materials.
Conclusion
Q&A: Solving Inequalities
Q: What is an inequality?
A: An inequality is a statement that compares two values using a mathematical symbol, such as <, >, ≤, or ≥.
Q: How do I write an inequality?
A: To write an inequality, you need to identify the variable and the value it is being compared to. For example, if you want to write an inequality that represents the number of kilometers Bill will bike, you would write:
x ≥ 4
This inequality states that the number of kilometers Bill will bike (x) is greater than or equal to 4.
Q: What is the difference between an inequality and an equation?
A: An equation is a statement that says two values are equal, using an equals sign (=). An inequality is a statement that compares two values using a mathematical symbol, such as <, >, ≤, or ≥.
Q: How do I solve an inequality?
A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. For example, if you have the inequality:
x + 2 ≥ 5
You can subtract 2 from both sides to get:
x ≥ 3
Q: What is the order of operations for solving inequalities?
A: The order of operations for solving inequalities is the same as for solving equations:
- Parentheses: Evaluate any expressions inside parentheses.
- Exponents: Evaluate any exponential expressions.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.
Q: How do I graph an inequality?
A: To graph an inequality, you need to draw a number line and mark the point that represents the value of the inequality. Then, you need to shade the region to the right or left of the point, depending on the direction of the inequality.
Q: What are some real-world applications of inequalities?
A: Inequalities have many real-world applications, such as:
- Finance: Inequalities can be used to model financial situations, such as investments and loans.
- Science: Inequalities can be used to model scientific situations, such as population growth and chemical reactions.
- Engineering: Inequalities can be used to model engineering situations, such as stress and strain on materials.
Q: How do I use inequalities in real-world problems?
A: To use inequalities in real-world problems, you need to identify the variable and the value it is being compared to. Then, you need to write an inequality that represents the situation and solve it to find the solution.
Q: What are some common mistakes to avoid when solving inequalities?
A: Some common mistakes to avoid when solving inequalities include:
- Not following the order of operations: Make sure to follow the order of operations when solving inequalities.
- Not isolating the variable: Make sure to isolate the variable on one side of the inequality sign.
- Not checking the solution: Make sure to check the solution to ensure it is correct.
Conclusion
In this article, we answered some common questions about solving inequalities. We covered topics such as writing inequalities, solving inequalities, graphing inequalities, and real-world applications of inequalities. We also discussed some common mistakes to avoid when solving inequalities.