Solve The Inequality:${ \begin{aligned} 12x + 17 - \left(6^2 - 30\right) \ \textless \ 47 \end{aligned} }$
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Introduction
In this article, we will delve into the world of inequalities and explore a specific problem that requires us to solve for the variable x. The given inequality is a complex expression that involves exponents, parentheses, and various mathematical operations. Our goal is to simplify the expression, isolate the variable, and find the solution set that satisfies the given inequality.
Understanding the Inequality
The given inequality is:
12x + 17 - (6^2 - 30) < 47
To begin solving this inequality, we need to follow the order of operations (PEMDAS):
- Evaluate the expression inside the parentheses: 6^2 - 30
- Simplify the expression: 36 - 30 = 6
- Rewrite the inequality with the simplified expression: 12x + 17 - 6 < 47
Simplifying the Inequality
Now that we have simplified the expression inside the parentheses, we can focus on simplifying the inequality further. We can start by combining like terms:
12x + 11 < 47
Isolating the Variable
Our goal is to isolate the variable x, so we need to get rid of the constant term on the left-hand side. We can do this by subtracting 11 from both sides of the inequality:
12x < 36
Solving for x
Now that we have isolated the variable x, we can solve for its value. To do this, we need to divide both sides of the inequality by 12:
x < 3
Conclusion
In this article, we have solved the given inequality by following the order of operations, simplifying the expression, isolating the variable, and finding the solution set. The final solution is x < 3, which means that any value of x that is less than 3 will satisfy the given inequality.
Tips and Tricks
When solving inequalities, it's essential to follow the order of operations and simplify the expression as much as possible. Additionally, be careful when dividing or multiplying both sides of the inequality by a negative number, as this can change the direction of the inequality.
Real-World Applications
Inequalities are used in various real-world applications, such as:
- Finance: Inequalities are used to calculate interest rates, investment returns, and other financial metrics.
- Science: Inequalities are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Common Mistakes
When solving inequalities, it's easy to make mistakes. Some common mistakes include:
- Failing to follow the order of operations
- Not simplifying the expression enough
- Dividing or multiplying both sides of the inequality by a negative number
- Not checking the solution set for extraneous solutions
Final Thoughts
Solving inequalities requires a combination of mathematical skills, logical thinking, and attention to detail. By following the order of operations, simplifying the expression, isolating the variable, and finding the solution set, we can solve even the most complex inequalities. Remember to be careful when dividing or multiplying both sides of the inequality by a negative number, and always check the solution set for extraneous solutions.
Additional Resources
For more information on solving inequalities, check out the following resources:
- Khan Academy: Inequalities
- Mathway: Inequality Solver
- Wolfram Alpha: Inequality Solver
Frequently Asked Questions
Q: What is the order of operations? A: The order of operations is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
Q: How do I simplify an expression? A: To simplify an expression, you need to combine like terms, eliminate any unnecessary parentheses, and perform any necessary calculations.
Q: What is the difference between an inequality and an equation? A: An inequality is a statement that two expressions are not equal, while an equation is a statement that two expressions are equal.
Q: How do I solve an inequality?
A: To solve an inequality, you need to follow the order of operations, simplify the expression, isolate the variable, and find the solution set.
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Introduction
In our previous article, we explored the world of inequalities and solved a complex expression. However, we know that solving inequalities can be a challenging task, and many students and professionals struggle to understand the concepts and techniques involved. In this article, we will provide a Q&A guide to help you better understand solving inequalities.
Q&A: Solving Inequalities
Q: What is the order of operations when solving inequalities?
A: The order of operations is the same as when solving equations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. This is often remembered using the acronym PEMDAS.
Q: How do I simplify an expression when solving an inequality?
A: To simplify an expression, you need to combine like terms, eliminate any unnecessary parentheses, and perform any necessary calculations. Remember to follow the order of operations to ensure that you are simplifying the expression correctly.
Q: What is the difference between an inequality and an equation?
A: An inequality is a statement that two expressions are not equal, while an equation is a statement that two expressions are equal. Inequalities often involve the use of symbols such as <, >, ≤, or ≥.
Q: How do I solve an inequality?
A: To solve an inequality, you need to follow the order of operations, simplify the expression, isolate the variable, and find the solution set. Remember to check the solution set for extraneous solutions.
Q: What is the solution set of an inequality?
A: The solution set of an inequality is the set of all values of the variable that satisfy the inequality. This can be a single value, a range of values, or even an empty set.
Q: How do I check for extraneous solutions?
A: To check for extraneous solutions, you need to plug the solution back into the original inequality and verify that it is true. If the solution is not true, then it is an extraneous solution and should be discarded.
Q: What are some common mistakes to avoid when solving inequalities?
A: Some common mistakes to avoid when solving inequalities include:
- Failing to follow the order of operations
- Not simplifying the expression enough
- Dividing or multiplying both sides of the inequality by a negative number
- Not checking the solution set for extraneous solutions
Q: How do I graph an inequality on a number line?
A: To graph an inequality on a number line, you need to plot a point on the number line that represents the solution set. If the inequality is of the form x < a, then you would plot a point to the left of a. If the inequality is of the form x > a, then you would plot a point to the right of a.
Q: What are some real-world applications of solving inequalities?
A: Solving inequalities has many real-world applications, including:
- Finance: Inequalities are used to calculate interest rates, investment returns, and other financial metrics.
- Science: Inequalities are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Conclusion
Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving even the most complex inequalities. Remember to follow the order of operations, simplify the expression, isolate the variable, and find the solution set. By avoiding common mistakes and checking for extraneous solutions, you can ensure that your solutions are accurate and reliable.
Additional Resources
For more information on solving inequalities, check out the following resources:
- Khan Academy: Inequalities
- Mathway: Inequality Solver
- Wolfram Alpha: Inequality Solver
Frequently Asked Questions
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 solve a quadratic inequality?
A: To solve a quadratic inequality, you need to factor the quadratic expression, if possible, and then use the factored form to find the solution set.
Q: What is the significance of the discriminant in solving quadratic inequalities?
A: The discriminant is a value that is used to determine the nature of the solutions to a quadratic equation. If the discriminant is positive, then the equation has two distinct real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then the equation has no real solutions.
Q: How do I graph a quadratic inequality on a number line?
A: To graph a quadratic inequality on a number line, you need to plot a point on the number line that represents the solution set. If the inequality is of the form x^2 + bx + c < d, then you would plot a point to the left or right of the vertex of the parabola.