Which Graph Shows The Solution To This Inequality?$\[ 3r + 2(12r + 7) \leq 5r - 8 \\]
Understanding the Basics of Inequalities
In mathematics, inequalities are used to compare two or more values. They are represented by symbols such as <, >, ≤, or ≥. In this article, we will focus on solving and graphing the solution to the inequality 3r + 2(12r + 7) ≤ 5r - 8. To begin, let's break down the inequality and understand its components.
Breaking Down the Inequality
The given inequality is 3r + 2(12r + 7) ≤ 5r - 8. To solve this inequality, we need to follow the order of operations (PEMDAS):
- Parentheses: Evaluate the expression inside the parentheses: 2(12r + 7) = 24r + 14.
- Exponents: None in this case.
- Multiplication and Division: Multiply and divide from left to right: 3r + 24r + 14 ≤ 5r - 8.
- Addition and Subtraction: Combine like terms: 27r + 14 ≤ 5r - 8.
Solving the Inequality
Now that we have simplified the inequality, let's solve for r:
27r + 14 ≤ 5r - 8
Subtract 5r from both sides:
22r + 14 ≤ -8
Subtract 14 from both sides:
22r ≤ -22
Divide both sides by 22:
r ≤ -1
Graphing the Solution
To graph the solution to the inequality, we need to represent the solution on a number line. The solution is r ≤ -1, which means that r is less than or equal to -1.
Graphing the Solution:
- Start by drawing a number line with a point at -1.
- Shade the region to the left of -1, including the point -1.
Understanding the Graph
The graph represents the solution to the inequality r ≤ -1. The shaded region indicates that all values of r less than or equal to -1 are part of the solution.
Visualizing the Graph
To visualize the graph, imagine a number line with a point at -1. The shaded region would be to the left of -1, including the point -1. This graph represents the solution to the inequality and helps us understand the values of r that satisfy the inequality.
Conclusion
In this article, we solved the inequality 3r + 2(12r + 7) ≤ 5r - 8 and graphed the solution on a number line. The solution is r ≤ -1, which means that r is less than or equal to -1. The graph represents the solution to the inequality and helps us understand the values of r that satisfy the inequality.
Real-World Applications
Solving inequalities and graphing the solution has many real-world applications. For example, in economics, inequalities can be used to model the relationship between two or more variables. In engineering, inequalities can be used to design and optimize systems. In finance, inequalities can be used to model the behavior of financial markets.
Common Mistakes to Avoid
When solving inequalities and graphing the solution, there are several common mistakes to avoid:
- Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.
- Not simplifying the inequality: Failing to simplify the inequality can make it difficult to solve and graph the solution.
- Not considering the direction of the inequality: Failing to consider the direction of the inequality can lead to incorrect solutions.
Tips and Tricks
When solving inequalities and graphing the solution, here are some tips and tricks to keep in mind:
- Use a number line: Using a number line can help you visualize the solution and make it easier to graph.
- Shade the region: Shading the region can help you see the solution more clearly.
- Check your work: Checking your work can help you catch any mistakes and ensure that your solution is correct.
Final Thoughts
Solving inequalities and graphing the solution is an important skill in mathematics. By following the steps outlined in this article, you can solve and graph the solution to any inequality. Remember to follow the order of operations, simplify the inequality, and consider the direction of the inequality. With practice and patience, you can become proficient in solving inequalities and graphing the solution.
Q: What is an inequality?
A: An inequality is a statement that compares two or more values using symbols such as <, >, ≤, or ≥.
Q: How do I solve an inequality?
A: To solve an inequality, follow the order of operations (PEMDAS) and simplify the inequality. Then, isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is an inequality that can be written in the form ax + b ≤ c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c ≤ d, where a, b, c, and d are constants.
Q: How do I graph the solution to an inequality?
A: To graph the solution to an inequality, draw a number line and shade the region that satisfies the inequality. If the inequality is of the form x ≤ a, shade the region to the left of a. If the inequality is of the form x ≥ a, shade the region to the right of a.
Q: What is the significance of the direction of the inequality?
A: The direction of the inequality is important because it determines the direction of the shading on the number line. If the inequality is of the form x ≤ a, the shading should be to the left of a. If the inequality is of the form x ≥ a, the shading should be to the right of a.
Q: Can I use a calculator to solve an inequality?
A: Yes, you can use a calculator to solve an inequality. However, it's always a good idea to check your work by hand to ensure that the solution is correct.
Q: How do I check my work when solving an inequality?
A: To check your work, plug the solution back into the original inequality and verify that it is true. If the solution is not true, recheck your work and try again.
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 (PEMDAS)
- Not simplifying the inequality
- Not considering the direction of the inequality
- Not checking your work
Q: How do I determine the solution to a compound inequality?
A: To determine the solution to a compound inequality, solve each inequality separately and then combine the solutions. For example, if the compound inequality is x > 2 and x < 5, the solution would be 2 < x < 5.
Q: Can I use a graphing calculator to graph the solution to an inequality?
A: Yes, you can use a graphing calculator to graph the solution to an inequality. However, it's always a good idea to check your work by hand to ensure that the solution is correct.
Q: How do I graph the solution to a system of inequalities?
A: To graph the solution to a system of inequalities, graph each inequality separately and then find the intersection of the two graphs. The intersection of the two graphs represents the solution to the system of inequalities.
Q: What are some real-world applications of solving inequalities?
A: Some real-world applications of solving inequalities include:
- Modeling the relationship between two or more variables in economics
- Designing and optimizing systems in engineering
- Modeling the behavior of financial markets in finance
Q: How do I determine the solution to a linear inequality with fractions?
A: To determine the solution to a linear inequality with fractions, multiply both sides of the inequality by the least common multiple (LCM) of the denominators to eliminate the fractions. Then, solve the resulting inequality.
Q: Can I use a computer program to solve an inequality?
A: Yes, you can use a computer program to solve an inequality. However, it's always a good idea to check your work by hand to ensure that the solution is correct.
Q: How do I graph the solution to a quadratic inequality?
A: To graph the solution to a quadratic inequality, graph the related quadratic function and then shade the region that satisfies the inequality. If the inequality is of the form x ≤ a, shade the region to the left of a. If the inequality is of the form x ≥ a, shade the region to the right of a.