Which Ordered Pair Makes Both Inequalities True?$\[ \begin{align*} y & \leq -x + 1 \\ y & \ \textgreater \ X \end{align*} \\]

Introduction

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.

Understanding the Inequalities

The given inequalities are:

• $y \leq -x + 1$
• $y > x$

The first inequality states that $y$ is less than or equal to $-x + 1$. This means that the value of $y$ is either less than or equal to the value of $-x + 1$. The second inequality states that $y$ is greater than $x$. This means that the value of $y$ is always greater than the value of $x$.

Graphing the Inequalities

To visualize the inequalities, we can graph them on a coordinate plane. The first inequality, $y \leq -x + 1$, can be graphed as a line with a slope of $-1$ and a y-intercept of $1$. The line is dashed, indicating that the inequality is not an equation. The region below the line represents the values of $y$ that satisfy the inequality.

The second inequality, $y > x$, can be graphed as a line with a slope of $1$ and a y-intercept of $0$. The line is dashed, indicating that the inequality is not an equation. The region above the line represents the values of $y$ that satisfy the inequality.

Finding the Intersection

To find the ordered pair that satisfies both inequalities, we need to find the intersection of the two regions. The intersection is the area where the two regions overlap.

To find the intersection, we can set the two inequalities equal to each other and solve for $x$ and $y$. However, since the inequalities are not equalities, we need to use a different approach.

Using a System of Inequalities

We can use a system of inequalities to find the intersection. A system of inequalities is a set of two or more inequalities that are solved simultaneously.

To solve the system of inequalities, we can use the following steps:

1. Graph the first inequality, $y \leq -x + 1$, on a coordinate plane.
2. Graph the second inequality, $y > x$, on the same coordinate plane.
3. Find the intersection of the two regions.
4. Check the intersection to make sure it satisfies both inequalities.

Solving the System of Inequalities

To solve the system of inequalities, we can use the following steps:

1. Graph the first inequality, $y \leq -x + 1$, on a coordinate plane.
2. Graph the second inequality, $y > x$, on the same coordinate plane.
3. Find the intersection of the two regions.

The intersection of the two regions is the area where the two lines intersect. To find the intersection, we can set the two equations equal to each other and solve for $x$ and $y$.

Finding the Ordered Pair

To find the ordered pair that satisfies both inequalities, we need to find the intersection of the two regions. The intersection is the area where the two regions overlap.

To find the intersection, we can use the following steps:

1. Set the two inequalities equal to each other and solve for $x$ and $y$.
2. Check the intersection to make sure it satisfies both inequalities.

Solving for x and y

To solve for $x$ and $y$, we can use the following steps:

1. Set the two inequalities equal to each other: $-x + 1 > x$
2. Simplify the inequality: $-2x + 1 > 0$
3. Solve for $x$: $-2x > -1$
4. Divide both sides by $-2$: $x < \frac{1}{2}$
5. Set the two inequalities equal to each other: $-x + 1 = x$
6. Simplify the equation: $-2x + 1 = 0$
7. Solve for $x$: $-2x = -1$
8. Divide both sides by $-2$: $x = \frac{1}{2}$
9. Substitute $x = \frac{1}{2}$ into one of the original inequalities to solve for $y$.

Finding the Value of y

To find the value of $y$, we can substitute $x = \frac{1}{2}$ into one of the original inequalities.

Substituting $x = \frac{1}{2}$ into the first inequality, $y \leq -x + 1$, we get:

$y \leq -\frac{1}{2} + 1$

Simplifying the inequality, we get:

$y \leq \frac{1}{2}$

Therefore, the value of $y$ is less than or equal to $\frac{1}{2}$.

Conclusion

In conclusion, to find the ordered pair that satisfies both inequalities, we need to find the intersection of the two regions. The intersection is the area where the two regions overlap.

To find the intersection, we can use a system of inequalities. A system of inequalities is a set of two or more inequalities that are solved simultaneously.

We can use the following steps to solve the system of inequalities:

1. Graph the first inequality, $y \leq -x + 1$, on a coordinate plane.
2. Graph the second inequality, $y > x$, on the same coordinate plane.
3. Find the intersection of the two regions.
4. Check the intersection to make sure it satisfies both inequalities.

By following these steps, we can find the ordered pair that satisfies both inequalities.

Final Answer

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

A: An inequality is a statement that compares two expressions using a mathematical symbol such as <, >, ≤, or ≥. An equation is a statement that says two expressions are equal.

Q: How do I graph an inequality on a coordinate plane?

A: To graph an inequality on a coordinate plane, you need to graph the related equation and then shade the region that satisfies the inequality. If the inequality is of the form y ≤ f(x), you shade the region below the graph of f(x). If the inequality is of the form y > f(x), you shade the region above the graph of f(x).

Q: How do I find the intersection of two inequalities?

A: To find the intersection of two inequalities, you need to solve the system of inequalities. This involves graphing the two inequalities on the same coordinate plane and finding the region where they overlap.

Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that are solved simultaneously. It involves finding the values of the variables that satisfy all the inequalities in the system.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to follow these steps:

1. Graph the first inequality on a coordinate plane.
2. Graph the second inequality on the same coordinate plane.
3. Find the intersection of the two regions.
4. Check the intersection to make sure it satisfies both inequalities.

Q: What is the difference between a linear inequality and a nonlinear inequality?

A: A linear inequality is an inequality that can be written in the form ax + by ≤ c, where a, b, and c are constants. A nonlinear inequality is an inequality that cannot be written in this form.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow these steps:

1. Graph the related equation on a coordinate plane.
2. Shade the region that satisfies the inequality.
3. Find the x-intercept and y-intercept of the related equation.
4. Use the intercepts to determine the direction of the inequality.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you need to follow these steps:

1. Graph the related equation on a coordinate plane.
2. Shade the region that satisfies the inequality.
3. Find the x-intercept and y-intercept of the related equation.
4. Use the intercepts to determine the direction of the inequality.

Q: What is the importance of solving inequalities?

A: Solving inequalities is important because it helps us to make decisions based on data. Inequalities are used in many real-world applications, such as finance, economics, and engineering.

Q: How do I apply inequalities in real-world situations?

A: To apply inequalities in real-world situations, you need to follow these steps:

1. Identify the variables and the relationships between them.
2. Write the inequalities that describe the relationships.
3. Solve the inequalities to find the values of the variables.
4. Use the solutions to make decisions or predictions.

Q: What are some common applications of inequalities?

A: Some common applications of inequalities include:

• Finance: Inequalities are used to calculate interest rates, investment returns, and credit scores.
• Economics: Inequalities are used to model economic systems, such as supply and demand curves.
• Engineering: Inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: How do I choose the right method for solving inequalities?

A: To choose the right method for solving inequalities, you need to consider the type of inequality, the number of variables, and the complexity of the problem. Some common methods for solving inequalities include:

• Graphing: This method involves graphing the related equation and shading the region that satisfies the inequality.
• Algebraic: This method involves solving the inequality using algebraic techniques, such as factoring and solving quadratic equations.
• Numerical: This method involves using numerical methods, such as the bisection method or the secant method, to approximate the solution.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

• Not considering the direction of the inequality.
• Not graphing the related equation.
• Not shading the region that satisfies the inequality.
• Not checking the solution to make sure it satisfies the inequality.

By following these tips and avoiding common mistakes, you can become proficient in solving inequalities and apply them to real-world situations.