Which Ordered Pair Makes Both Inequalities True?${ \begin{align*} y & \ \textgreater \ -2x + 3 \ y & \leq X - 2 \end{align*} }$A. (0, 0) B. (0, -1) C. (1, 1)

Solving Inequalities: Finding the Ordered Pair that Satisfies Both Conditions

In mathematics, inequalities are used to describe relationships between variables. When solving inequalities, we often need to find the values of the variables that satisfy both inequalities simultaneously. In this article, we will explore how to solve inequalities and find the ordered pair that makes both inequalities true.

An inequality is a statement that compares two expressions using a mathematical symbol, such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). Inequalities can be used to describe relationships between variables, and they are often used in mathematical modeling and problem-solving.

We are given two inequalities:

1. $y > -2x + 3$
2. $y \leq x - 2$

Our goal is to find the ordered pair $(x, y)$ that satisfies both inequalities.

To solve the first inequality, we can start by isolating the variable $y$. We can do this by subtracting $-2x + 3$ from both sides of the inequality:

$$y - (-2x + 3) > 0$$

Simplifying the left-hand side, we get:

$$y + 2x - 3 > 0$$

Now, we can add $3$ to both sides of the inequality to get:

$$y + 2x > 3$$

To solve the second inequality, we can start by isolating the variable $y$. We can do this by subtracting $x - 2$ from both sides of the inequality:

$$y - (x - 2) \leq 0$$

Simplifying the left-hand side, we get:

$$y - x + 2 \leq 0$$

Now, we can add $x$ to both sides of the inequality to get:

$$y + 2 \leq x$$

Now that we have solved both inequalities, we can find the ordered pair $(x, y)$ that satisfies both inequalities. To do this, we can graph the two inequalities on a coordinate plane and find the intersection point.

To graph the first inequality, we can use the following steps:

1. Plot the line $y = -2x + 3$ on the coordinate plane.
2. Shade the region above the line, since the inequality is $y > -2x + 3$.

To graph the second inequality, we can use the following steps:

1. Plot the line $y = x - 2$ on the coordinate plane.
2. Shade the region below the line, since the inequality is $y \leq x - 2$.

The intersection point of the two lines is the point where the two inequalities are satisfied simultaneously. To find the intersection point, we can set the two equations equal to each other and solve for $x$ and $y$.

Setting the two equations equal to each other, we get:

$$-2x + 3 = x - 2$$

Solving for $x$, we get:

$$-3x = -5$$

Dividing both sides by $-3$, we get:

$$x = \frac{5}{3}$$

Now, we can substitute $x = \frac{5}{3}$ into one of the original equations to solve for $y$. Substituting $x = \frac{5}{3}$ into the first equation, we get:

$$y > -2\left(\frac{5}{3}\right) + 3$$

Simplifying the right-hand side, we get:

$$y > -\frac{10}{3} + 3$$

Adding $\frac{10}{3}$ to both sides, we get:

$$y > \frac{1}{3}$$

Now, we can substitute $x = \frac{5}{3}$ into the second equation to solve for $y$. Substituting $x = \frac{5}{3}$ into the second equation, we get:

$$y \leq \frac{5}{3} - 2$$

Simplifying the right-hand side, we get:

$$y \leq \frac{1}{3}$$

Since the two inequalities are satisfied simultaneously, we can conclude that the ordered pair $(x, y)$ that satisfies both inequalities is:

$$(x, y) = \left(\frac{5}{3}, \frac{1}{3}\right)$$

In this article, we have explored how to solve inequalities and find the ordered pair that makes both inequalities true. We have used the given inequalities $y > -2x + 3$ and $y \leq x - 2$ to find the intersection point of the two lines, which is the point where the two inequalities are satisfied simultaneously. We have shown that the ordered pair $(x, y)$ that satisfies both inequalities is $(x, y) = \left(\frac{5}{3}, \frac{1}{3}\right)$.

The correct answer is:

• $\boxed{\left(\frac{5}{3}, \frac{1}{3}\right)}$
  Frequently Asked Questions (FAQs) about Solving Inequalities

A: An inequality is a statement that compares two expressions using a mathematical symbol, such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).

A: To solve an inequality, you can start by isolating the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

A: A linear inequality is an inequality that can be written in the form $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

A: To graph an inequality on a coordinate plane, you can start by plotting the line that represents the boundary of the inequality. Then, you can shade the region above or below the line, depending on whether the inequality is greater than or less than the boundary.

A: The intersection point of two lines is the point where the two lines intersect. This point represents the solution to the system of equations formed by the two lines.

A: To find the intersection point of two lines, you can set the two equations equal to each other and solve for the variables. This will give you the coordinates of the intersection point.

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. A system of linear inequalities is a set of two or more linear inequalities that are solved simultaneously.

A: To solve a system of linear inequalities, you can start by graphing the inequalities on a coordinate plane. Then, you can find the intersection point of the two lines, which represents the solution to the system of inequalities.

A: The solution to a system of linear inequalities is the set of all points that satisfy both inequalities simultaneously. This set of points represents the solution to the system of inequalities.

A: To check your solution to a system of linear inequalities, you can substitute the coordinates of the solution into both inequalities and check that they are true. If the solution satisfies both inequalities, then it is the correct solution.

A: Some common mistakes to avoid when solving systems of linear inequalities include:

• Graphing the inequalities incorrectly
• Finding the intersection point incorrectly
• Not checking the solution in both inequalities
• Not considering all possible solutions

A: You can practice solving systems of linear inequalities by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear inequalities on your own, using a graphing calculator or other tool to help you visualize the solution.

A: Solving systems of linear inequalities has many real-world applications, including:

• Optimization problems, such as finding the maximum or minimum value of a function
• Scheduling problems, such as finding the best schedule for a set of tasks
• Resource allocation problems, such as finding the best way to allocate resources to a set of projects
• Network flow problems, such as finding the best way to route traffic through a network.