Solve For $y$. Reduce Any Fractions To Their Lowest Terms. Do Not Round Your Answer Or Use Mixed Fractions.$5y + 3 \ \textgreater \ -7y + 13$
Introduction to Solving Linear Inequalities
Solving linear inequalities is a fundamental concept in mathematics, and it plays a crucial role in various fields such as algebra, geometry, and calculus. In this article, we will focus on solving a linear inequality of the form $5y + 3 \ \textgreater \ -7y + 13$. Our goal is to isolate the variable $y$ and reduce any fractions to their lowest terms.
Understanding the Inequality
Before we start solving the inequality, let's understand the concept of inequalities. An inequality is a statement that compares two expressions using a mathematical symbol such as $\textgreater$, $\textless$, $\textgreater \textgreater$, or $\textless \textless$. In this case, we have the inequality $5y + 3 \ \textgreater \ -7y + 13$, which means that the expression $5y + 3$ is greater than the expression $-7y + 13$.
Isolating the Variable $y$
To solve the inequality, we need to isolate the variable $y$. We can do this by adding or subtracting the same value to both sides of the inequality. In this case, we can add $7y$ to both sides of the inequality to get:
Simplifying the left-hand side of the inequality, we get:
Subtracting 3 from Both Sides
Next, we can subtract 3 from both sides of the inequality to get:
Simplifying the left-hand side of the inequality, we get:
Dividing Both Sides by 12
Finally, we can divide both sides of the inequality by 12 to get:
Simplifying the left-hand side of the inequality, we get:
Conclusion
In conclusion, we have solved the linear inequality $5y + 3 \ \textgreater \ -7y + 13$ and isolated the variable $y$. We have reduced any fractions to their lowest terms and obtained the solution $y \ \textgreater \ \frac{5}{6}$. This solution means that the value of $y$ must be greater than $\frac{5}{6}$.
Final Answer
The final answer is $y \ \textgreater \ \frac{5}{6}$.
Frequently Asked Questions
Q: What is the solution to the inequality $5y + 3 \ \textgreater \ -7y + 13$?
A: The solution to the inequality is $y \ \textgreater \ \frac{5}{6}$.
Q: How do I reduce fractions to their lowest terms?
A: To reduce fractions to their lowest terms, you can divide both the numerator and the denominator by their greatest common divisor (GCD).
Q: What is the difference between a linear inequality and a linear equation?
A: A linear inequality is a statement that compares two expressions using a mathematical symbol such as $\textgreater$, $\textless$, $\textgreater \textgreater$, or $\textless \textless$. A linear equation, on the other hand, is a statement that equates two expressions using an equals sign.
Step-by-Step Solution
Step 1: Add 7y to Both Sides
Add 7y to both sides of the inequality to get:
Step 2: Simplify the Left-Hand Side
Simplifying the left-hand side of the inequality, we get:
Step 3: Subtract 3 from Both Sides
Subtract 3 from both sides of the inequality to get:
Step 4: Simplify the Left-Hand Side
Simplifying the left-hand side of the inequality, we get:
Step 5: Divide Both Sides by 12
Divide both sides of the inequality by 12 to get:
Step 6: Simplify the Left-Hand Side
Simplifying the left-hand side of the inequality, we get:
Related Topics
- Solving Linear Equations
- Solving Quadratic Equations
- Graphing Linear Inequalities
- Solving Systems of Linear Equations
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "College Algebra" by James Stewart
- [3] "Linear Algebra and Its Applications" by Gilbert Strang
Keywords
- Linear Inequality
- Solving Linear Inequalities
- Isolating the Variable
- Reducing Fractions to Their Lowest Terms
- Greatest Common Divisor (GCD)
- Linear Equation
- Graphing Linear Inequalities
- Systems of Linear Equations
Introduction
Solving linear inequalities is a fundamental concept in mathematics, and it plays a crucial role in various fields such as algebra, geometry, and calculus. In this article, we will provide a Q&A section to help you understand and solve linear inequalities.
Q: What is a linear inequality?
A: A linear inequality is a statement that compares two expressions using a mathematical symbol such as $\textgreater$, $\textless$, $\textgreater \textgreater$, or $\textless \textless$.
Q: How do I solve a linear inequality?
A: To solve a linear inequality, you need to isolate the variable by adding or subtracting the same value to both sides of the inequality.
Q: What is the difference between a linear inequality and a linear equation?
A: A linear inequality is a statement that compares two expressions using a mathematical symbol such as $\textgreater$, $\textless$, $\textgreater \textgreater$, or $\textless \textless$. A linear equation, on the other hand, is a statement that equates two expressions using an equals sign.
Q: How do I reduce fractions to their lowest terms?
A: To reduce fractions to their lowest terms, you can divide both the numerator and the denominator by their greatest common divisor (GCD).
Q: What is the greatest common divisor (GCD)?
A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.
Q: How do I graph a linear inequality?
A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality.
Q: What is the solution to the inequality $5y + 3 \ \textgreater \ -7y + 13$?
A: The solution to the inequality is $y \ \textgreater \ \frac{5}{6}$.
Q: How do I solve a system of linear inequalities?
A: To solve a system of linear inequalities, you need to find the solution that satisfies all the inequalities.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is a statement that compares two expressions using a mathematical symbol such as $\textgreater$, $\textless$, $\textgreater \textgreater$, or $\textless \textless$. A quadratic inequality, on the other hand, is a statement that compares two quadratic expressions using a mathematical symbol such as $\textgreater$, $\textless$, $\textgreater \textgreater$, or $\textless \textless$.
Q: How do I solve a quadratic inequality?
A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution.
Q: What is the solution to the inequality $x^2 + 4x + 4 \ \textless \ 0$?
A: The solution to the inequality is $x \ \textless \ -2$.
Q: How do I graph a quadratic inequality?
A: To graph a quadratic inequality, you need to graph the corresponding quadratic equation and then shade the region that satisfies the inequality.
Q: What is the difference between a linear inequality and a rational inequality?
A: A linear inequality is a statement that compares two expressions using a mathematical symbol such as $\textgreater$, $\textless$, $\textgreater \textgreater$, or $\textless \textless$. A rational inequality, on the other hand, is a statement that compares two rational expressions using a mathematical symbol such as $\textgreater$, $\textless$, $\textgreater \textgreater$, or $\textless \textless$.
Q: How do I solve a rational inequality?
A: To solve a rational inequality, you need to factor the numerator and denominator and then use the sign of the rational expression to determine the solution.
Q: What is the solution to the inequality $\frac{x + 2}{x - 1} \ \textgreater \ 0$?
A: The solution to the inequality is $x \ \textless \ -2$ or $x \ \textgreater \ 1$.
Q: How do I graph a rational inequality?
A: To graph a rational inequality, you need to graph the corresponding rational equation and then shade the region that satisfies the inequality.
Q&A: Solving Linear Inequalities
Q: What is the solution to the inequality $3x - 2 \ \textless \ 5$?
A: The solution to the inequality is $x \ \textless \ \frac{7}{3}$.
Q: How do I solve the inequality $2x + 5 \ \textgreater \ 3x - 2$?
A: To solve the inequality, you need to isolate the variable by subtracting 2x from both sides of the inequality.
Q: What is the solution to the inequality $x^2 - 4 \ \textless \ 0$?
A: The solution to the inequality is $-2 \ \textless \ x \ \textless \ 2$.
Q: How do I solve the inequality $\frac{x + 1}{x - 2} \ \textgreater \ 0$?
A: To solve the inequality, you need to factor the numerator and denominator and then use the sign of the rational expression to determine the solution.
Q: What is the solution to the inequality $x^2 + 2x - 3 \ \textless \ 0$?
A: The solution to the inequality is $-3 \ \textless \ x \ \textless \ 1$.
Q: How do I solve the inequality $x^2 - 4x + 4 \ \textgreater \ 0$?
A: To solve the inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution.
Q: What is the solution to the inequality $\frac{x - 1}{x + 2} \ \textless \ 0$?
A: The solution to the inequality is $x \ \textless \ -2$ or $x \ \textgreater \ 1$.
Q: How do I solve the inequality $x^2 + 4x + 4 \ \textgreater \ 0$?
A: To solve the inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution.
Q: What is the solution to the inequality $x^2 - 2x - 3 \ \textless \ 0$?
A: The solution to the inequality is $-1 \ \textless \ x \ \textless \ 3$.
Q: How do I solve the inequality $\frac{x + 2}{x - 1} \ \textless \ 0$?
A: To solve the inequality, you need to factor the numerator and denominator and then use the sign of the rational expression to determine the solution.
Q: What is the solution to the inequality $x^2 + 2x - 3 \ \textgreater \ 0$?
A: The solution to the inequality is $-3 \ \textless \ x \ \textless \ 1$.
Q: How do I solve the inequality $x^2 - 4x + 4 \ \textless \ 0$?
A: To solve the inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution.
Q: What is the solution to the inequality $\frac{x - 1}{x + 2} \ \textgreater \ 0$?
A: The solution to the inequality is $x \ \textless \ -2$ or $x \ \textgreater \ 1$.
Q: How do I solve the inequality $x^2 + 4x + 4 \ \textless \ 0$?
A: To solve the inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution.
Q: What is the solution to the inequality $x^2 - 2x - 3 \ \textgreater \ 0$?
A: The solution to the inequality is $-1 \ \textless \ x \ \textless \ 3$.
Q: How do I solve the inequality $\frac{x + 2}{x - 1} \ \textless \ 0$?
A: To solve the inequality, you need to factor the numerator and denominator and then use the sign of the rational expression to determine the solution.
Conclusion
In conclusion, solving linear inequalities is a fundamental concept in mathematics, and it plays a crucial role in various fields such as algebra, geometry, and calculus. In this article, we have provided a Q&A section to help you understand and solve linear inequalities. We have also provided examples and solutions to various linear inequalities.