Which Ordered Pair Makes Both Inequalities True?${ \begin{array}{l} y \ \textgreater \ -2x + 3 \ y \leq X - 2 \end{array} }$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 we have two inequalities, we need to find the values of the variables that satisfy both conditions. 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, indicating whether one is greater than, less than, or equal to the other. Inequalities can be written in various forms, including:

• Greater than (>): $y > -2x + 3$
• Less than or equal to (≤): $y \leq x - 2$

The first inequality is $y > -2x + 3$. To solve this inequality, we need to isolate the variable $y$. We can do this by subtracting $-2x + 3$ from both sides of the inequality.

$$y > -2x + 3$$

Subtracting $-2x + 3$ from both sides gives us:

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

Simplifying the left-hand side, we get:

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

Now, we can rewrite the inequality in slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

$$y > -2x + 3$$

The slope of this line is $-2$, and the y-intercept is $3$. This means that the line has a negative slope and intersects the y-axis at the point $(0, 3)$.

The second inequality is $y \leq x - 2$. To solve this inequality, we need to isolate the variable $y$. We can do this by subtracting $x - 2$ from both sides of the inequality.

$$y \leq x - 2$$

Subtracting $x - 2$ from both sides gives us:

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

Simplifying the left-hand side, we get:

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

Now, we can rewrite the inequality in slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

$$y \leq x - 2$$

The slope of this line is $1$, and the y-intercept is $-2$. This means that the line has a positive slope and intersects the y-axis at the point $(0, -2)$.

Now that we have solved both inequalities, we need to find the ordered pair that satisfies both conditions. To do this, we can graph the two lines on a coordinate plane and find the point of intersection.

The first line has a slope of $-2$ and a y-intercept of $3$, while the second line has a slope of $1$ and a y-intercept of $-2$. We can graph these lines on a coordinate plane and find the point of intersection.

To graph the first line, we can use the slope-intercept form of the equation, which is $y = mx + b$. We can plug in the values of the slope and y-intercept to get:

$$y = -2x + 3$$

We can graph this line on a coordinate plane by plotting the y-intercept at the point $(0, 3)$ and then drawing a line with a slope of $-2$.

To graph the second line, we can use the slope-intercept form of the equation, which is $y = mx + b$. We can plug in the values of the slope and y-intercept to get:

$$y = x - 2$$

We can graph this line on a coordinate plane by plotting the y-intercept at the point $(0, -2)$ and then drawing a line with a slope of $1$.

Now that we have graphed both lines, we can find the point of intersection by finding the point where the two lines cross. We can do this by setting the two equations equal to each other and solving for $x$.

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

Subtracting $x$ from both sides gives us:

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

Subtracting $3$ from both sides gives us:

$$-3x = -5$$

Dividing both sides by $-3$ gives us:

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

Now that we have found the value of $x$, we can substitute it into one of the original equations to find the value of $y$. We can use the first equation, which is $y > -2x + 3$.

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

Simplifying the right-hand side, we get:

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

Adding $\frac{10}{3}$ to both sides gives us:

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

Now that we have found the value of $y$, we can write the ordered pair that satisfies both inequalities.

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

In this article, we have explored how to solve inequalities and find the ordered pair that satisfies both conditions. We have used the slope-intercept form of the equation to graph the two lines on a coordinate plane and found the point of intersection. We have then used the point of intersection to find the ordered pair that satisfies both inequalities.

The ordered pair that makes both inequalities true is:

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

This is the correct answer.
Q&A: Solving Inequalities and Finding the Ordered Pair

In our previous article, we explored how to solve inequalities and find the ordered pair that satisfies both conditions. We used the slope-intercept form of the equation to graph the two lines on a coordinate plane and found the point of intersection. In this article, we will answer some frequently asked questions about solving inequalities and finding the ordered pair.

A: A linear inequality is an inequality that can be written in the form $y > mx + b$ or $y < mx + b$, where $m$ is the slope and $b$ is the y-intercept. A linear equation, on the other hand, is an equation that can be written in the form $y = mx + b$. The main difference between a linear inequality and a linear equation is that a linear inequality has a greater than or less than symbol, while a linear equation has an equal to symbol.

A: To graph a linear inequality on a coordinate plane, you can use the slope-intercept form of the equation, which is $y = mx + b$. You can plot the y-intercept at the point $(0, b)$ and then draw a line with a slope of $m$. If the inequality is greater than or less than, you can shade the region above or below the line.

A: To find the point of intersection between two linear inequalities, you can set the two equations equal to each other and solve for $x$. You can then substitute the value of $x$ into one of the original equations to find the value of $y$.

A: The point of intersection is significant in solving inequalities because it represents the point where the two lines cross. This point is the solution to the system of inequalities and represents the ordered pair that satisfies both conditions.

A: No, you cannot use the point of intersection to find the solution to a system of inequalities with more than two variables. The point of intersection is only applicable to systems of inequalities with two variables.

A: To determine if a point is a solution to a system of inequalities, you can substitute the values of $x$ and $y$ into both inequalities and check if the point satisfies both conditions.

A: Yes, you can use technology to solve systems of inequalities. Graphing calculators and computer software can be used to graph the lines and find the point of intersection.

In this article, we have answered some frequently asked questions about solving inequalities and finding the ordered pair. We have discussed the difference between a linear inequality and a linear equation, how to graph a linear inequality on a coordinate plane, and how to find the point of intersection between two linear inequalities. We have also discussed the significance of the point of intersection and how to determine if a point is a solution to a system of inequalities.

• Graphing Inequalities
• Solving Systems of Inequalities
• Graphing Calculator Software

Solving inequalities and finding the ordered pair is an important concept in mathematics. By understanding how to solve inequalities and find the ordered pair, you can apply this knowledge to real-world problems and make informed decisions. Remember to always use the slope-intercept form of the equation to graph the lines and find the point of intersection.