Understanding Solutions to Inequalities

Oct 13, 2025 by miodeya.com 33 views

Introduction

In mathematics, inequalities are used to compare values and determine the relationship between them. One common type of inequality is the linear inequality, which involves a linear expression and a constant. In this article, we will determine if the ordered pair $(6,4)$ is a solution to the inequality $y < \frac{3}{4}x - 3$ . To do this, we need to understand the concept of a solution to an inequality and how to graph the related linear equation.

Understanding Solutions to Inequalities

A solution to an inequality is a value or a set of values that make the inequality true. In the case of the inequality $y < \frac{3}{4}x - 3$ , any value of $y$ that is less than the value of $\frac{3}{4}x - 3$ is a solution. This means that we need to find the values of $x$ and $y$ that satisfy the inequality.

Graphing the Linear Equation

To graph the linear equation $y = \frac{3}{4}x - 3$ , we need to find two points on the line. We can do this by substituting different values of $x$ into the equation and solving for $y$ . Let's find the points $(0, -3)$ and $(4, 0)$ .

For the point $(0, -3)$ , we substitute $x = 0$ into the equation: ${y = \frac{3}{4}(0) - 3}$ ${y = -3}$
For the point $(4, 0)$ , we substitute $x = 4$ into the equation: ${y = \frac{3}{4}(4) - 3}$ ${y = 0}$

Now that we have two points on the line, we can graph the line by drawing a line through the points $(0, -3)$ and $(4, 0)$ .

Determining if the Ordered Pair is a Solution

To determine if the ordered pair $(6,4)$ is a solution to the inequality $y < \frac{3}{4}x - 3$ , we need to substitute the values of $x$ and $y$ into the inequality and check if it is true.

Substitute $x = 6$ and $y = 4$ into the inequality: ${4 < \frac{3}{4}(6) - 3}$ ${4 < 4.5 - 3}$ ${4 < 1.5}$

Since $4$ is not less than $1.5$ , the ordered pair $(6,4)$ is not a solution to the inequality.

Conclusion

In conclusion, the ordered pair $(6,4)$ is not a solution to the inequality $y < \frac{3}{4}x - 3$ . This is because the value of $y$ is not less than the value of $\frac{3}{4}x - 3$ when $x = 6$ . Therefore, the correct answer is:

C. No, because $(6,4)$ is not below the line.

Discussion

This problem requires the student to understand the concept of a solution to an inequality and how to graph the related linear equation. The student must also be able to substitute values into the inequality and check if it is true. This problem is a good example of how to use algebraic methods to solve inequalities and how to graph linear equations.

Key Concepts

Solutions to inequalities
Graphing linear equations
Substituting values into inequalities
Checking if an ordered pair is a solution to an inequality

Practice Problems

Determine if the ordered pair $(2,3)$ is a solution to the inequality $y < 2x - 1$ .
Graph the linear equation $y = \frac{1}{2}x + 2$ and determine if the ordered pair $(4,3)$ is a solution to the inequality $y < \frac{1}{2}x + 2$ .
Determine if the ordered pair $(5,2)$ is a solution to the inequality $y > 2x - 3$ .

Solutions to Practice Problems

To determine if the ordered pair $(2,3)$ is a solution to the inequality $y < 2x - 1$ , we need to substitute the values of $x$ and $y$ into the inequality and check if it is true.

Substitute $x = 2$ and $y = 3$ into the inequality: ${3 < 2(2) - 1}$ ${3 < 4 - 1}$ ${3 < 3}$

Since $3$ is not less than $3$ , the ordered pair $(2,3)$ is not a solution to the inequality.

To graph the linear equation $y = \frac{1}{2}x + 2$ , we need to find two points on the line. We can do this by substituting different values of $x$ into the equation and solving for $y$ . Let's find the points $(0, 2)$ and $(4, 3)$ .

For the point $(0, 2)$ , we substitute $x = 0$ into the equation: ${y = \frac{1}{2}(0) + 2}$ ${y = 2}$
For the point $(4, 3)$ , we substitute $x = 4$ into the equation: ${y = \frac{1}{2}(4) + 2}$ ${y = 3}$

Now that we have two points on the line, we can graph the line by drawing a line through the points $(0, 2)$ and $(4, 3)$ .

To determine if the ordered pair $(4,3)$ is a solution to the inequality $y < \frac{1}{2}x + 2$ , we need to substitute the values of $x$ and $y$ into the inequality and check if it is true.

Substitute $x = 4$ and $y = 3$ into the inequality: ${3 < \frac{1}{2}(4) + 2}$ ${3 < 2 + 2}$ ${3 < 4}$

Since $3$ is less than $4$ , the ordered pair $(4,3)$ is a solution to the inequality.

To determine if the ordered pair $(5,2)$ is a solution to the inequality $y > 2x - 3$ , we need to substitute the values of $x$ and $y$ into the inequality and check if it is true.

Substitute $x = 5$ and $y = 2$ into the inequality: ${2 > 2(5) - 3}$ ${2 > 10 - 3}$ ${2 > 7}$

Since $2$ is not greater than $7$ , the ordered pair $(5,2)$ is not a solution to the inequality.

Introduction

In the previous article, we discussed how to determine if an ordered pair is a solution to an inequality. In this article, we will answer some frequently asked questions about solving inequalities and provide additional examples to help you understand the concept.

Q&A

Q: What is the difference between a solution to an inequality and a solution to an equation?

A: A solution to an equation is a value that makes the equation true, while a solution to an inequality is a value that makes the inequality true. In other words, a solution to an equation is a value that satisfies the equation, while a solution to an inequality is a value that satisfies the inequality.

Q: How do I determine if an ordered pair is a solution to an inequality?

A: To determine if an ordered pair is a solution to an inequality, you need to substitute the values of x and y into the inequality and check if it is true. If the inequality is true, then the ordered pair is a solution to the inequality.

Q: What if the inequality is a compound inequality, such as $y < 2x - 1$ or $y > 2x - 1$ ?

A: To determine if an ordered pair is a solution to a compound inequality, you need to substitute the values of x and y into the inequality and check if it is true. If the inequality is true, then the ordered pair is a solution to the inequality.

Q: How do I graph a linear equation and determine if an ordered pair is a solution to the inequality?

A: To graph a linear equation, you need to find two points on the line and draw a line through the points. To determine if an ordered pair is a solution to the inequality, you need to substitute the values of x and y into the inequality and check if it is true.

Q: What if the inequality is a quadratic inequality, such as $y < x^2 - 4$ ?

A: To determine if an ordered pair is a solution to a quadratic inequality, you need to substitute the values of x and y into the inequality and check if it is true. You may also need to graph the related quadratic equation and determine if the ordered pair is a solution to the inequality.

Q: How do I determine if an ordered pair is a solution to a system of inequalities?

A: To determine if an ordered pair is a solution to a system of inequalities, you need to substitute the values of x and y into each inequality and check if it is true. If all the inequalities are true, then the ordered pair is a solution to the system of inequalities.

Examples

Example 1: Determining if an ordered pair is a solution to an inequality

Determine if the ordered pair $(2,3)$ is a solution to the inequality $y < 2x - 1$ .

Substitute $x = 2$ and $y = 3$ into the inequality: ${3 < 2(2) - 1}$ ${3 < 4 - 1}$ ${3 < 3}$

Since $3$ is not less than $3$ , the ordered pair $(2,3)$ is not a solution to the inequality.

Example 2: Graphing a linear equation and determining if an ordered pair is a solution to the inequality

Graph the linear equation $y = 2x - 1$ and determine if the ordered pair $(4,3)$ is a solution to the inequality $y < 2x - 1$ .

Find two points on the line by substituting different values of x into the equation and solving for y. Let's find the points $(0, -1)$ and $(4, 7)$ .
For the point $(0, -1)$ , we substitute $x = 0$ into the equation: ${y = 2(0) - 1}$ ${y = -1}$
For the point $(4, 7)$ , we substitute $x = 4$ into the equation: ${y = 2(4) - 1}$ ${y = 7}$
Now that we have two points on the line, we can graph the line by drawing a line through the points $(0, -1)$ and $(4, 7)$ .
To determine if the ordered pair $(4,3)$ is a solution to the inequality $y < 2x - 1$ , we need to substitute the values of x and y into the inequality and check if it is true.
Substitute $x = 4$ and $y = 3$ into the inequality: ${3 < 2(4) - 1}$ ${3 < 8 - 1}$ ${3 < 7}$

Since $3$ is less than $7$ , the ordered pair $(4,3)$ is a solution to the inequality.

Example 3: Determining if an ordered pair is a solution to a system of inequalities

Determine if the ordered pair $(5,2)$ is a solution to the system of inequalities $y < 2x - 1$ and $y > 2x - 3$ .

Substitute $x = 5$ and $y = 2$ into the first inequality: ${2 < 2(5) - 1}$ ${2 < 10 - 1}$ ${2 < 9}$

Since $2$ is less than $9$ , the ordered pair $(5,2)$ is a solution to the first inequality.

Substitute $x = 5$ and $y = 2$ into the second inequality: ${2 > 2(5) - 3}$ ${2 > 10 - 3}$ ${2 > 7}$

Since $2$ is not greater than $7$ , the ordered pair $(5,2)$ is not a solution to the second inequality.

Since the ordered pair $(5,2)$ is not a solution to both inequalities, it is not a solution to the system of inequalities.

Conclusion

In conclusion, determining if an ordered pair is a solution to an inequality requires substituting the values of x and y into the inequality and checking if it is true. You may also need to graph the related linear equation and determine if the ordered pair is a solution to the inequality. Additionally, you may need to determine if an ordered pair is a solution to a system of inequalities by substituting the values of x and y into each inequality and checking if it is true.

Key Concepts

Solutions to inequalities
Graphing linear equations
Substituting values into inequalities
Checking if an ordered pair is a solution to an inequality
Determining if an ordered pair is a solution to a system of inequalities

Practice Problems

Determine if the ordered pair $(3,4)$ is a solution to the inequality $y < 2x - 1$ .
Graph the linear equation $y = 2x - 1$ and determine if the ordered pair $(4,3)$ is a solution to the inequality $y < 2x - 1$ .
Determine if the ordered pair $(5,2)$ is a solution to the system of inequalities $y < 2x - 1$ and $y > 2x - 3$ .

Solutions to Practice Problems

To determine if the ordered pair $(3,4)$ is a solution to the inequality $y < 2x - 1$ , we need to substitute the values of x and y into the inequality and check if it is true.

Substitute $x = 3$ and $y = 4$ into the inequality: ${4 < 2(3) - 1}$ ${4 < 6 - 1}$ ${4 < 5}$

Since $4$ is not less than $5$ , the ordered pair $(3,4)$ is not a solution to the inequality.

To graph the linear equation $y = 2x - 1$ , we need to find two points on the line. We can do this by substituting different values of x into the equation and solving for y. Let's find the points $(0, -1)$ and $(4, 7)$ .

For the point $(0, -1)$ , we substitute $x = 0$ into the equation: ${y = 2(0) - 1}$ ${y = -1}$
For the point $(4, 7)$ , we substitute $x = 4$ into the equation: ${y = 2(4) - 1}$ ${y = 7}$
Now that we have two points on the line, we can graph the line by drawing a line through the points $(0, -1)$ and $(4, 7)$ .
To determine if the ordered pair $(4,3)$ is a solution to the inequality $y < 2x - 1$ , we need to substitute the values of x and y into the inequality and check if it is true.
Substitute $x = 4$ and $y = 3$ into the inequality: ${3 < 2(4) - 1}$ ${3 < 8 - 1}$ ${3 < 7}$

Since $3$ is less than $7$ , the ordered pair $(4,3)$ is a solution to the inequality.

To determine if the ordered pair $(5,2)$ is a solution to the system of inequalities $y < 2x - 1$ and $y > 2x - 3$ , we need to substitute the values of x and y into each inequality and check if it is true.

Substitute $x = 5$ and $