Unit 2 linear functions homework 7 linear inequalities – Embark on an intellectual journey with Unit 2 Linear Functions Homework 7: Linear Inequalities. This exploration unveils the fascinating realm of linear inequalities, where the boundaries of linear functions are tested and extended. Prepare to unravel the intricacies of graphing, solving, and applying these mathematical tools to real-world scenarios.

Delve into the fundamental concepts of linear inequalities, mastering their definition and properties. Discover the art of graphing these inequalities, exploring the different types and their corresponding regions. Equip yourself with a comprehensive understanding of solving linear inequalities, employing various methods to find their solutions.

1. Linear Functions Overview: Unit 2 Linear Functions Homework 7 Linear Inequalities

A linear function is a function that has a constant rate of change. This means that the output of a linear function increases or decreases by the same amount for each unit increase or decrease in the input. Linear functions are often used to model real-world phenomena, such as the growth of a population or the cost of a product.

The equation of a linear function is y = mx + b, where m is the slope of the function and b is the y-intercept. The slope of a linear function is the rate of change of the function, and the y-intercept is the value of the function when the input is 0.

Linear functions can be graphed by plotting the points (x, y) that satisfy the equation of the function. The graph of a linear function is a straight line.

2. Linear Inequalities

A linear inequality is an inequality that can be written in the form Ax + B > C, Ax + B < C, Ax + B ≥ C, or Ax + B ≤ C, where A, B, and C are real numbers.

The solution to a linear inequality is the set of all real numbers that make the inequality true.

Linear inequalities can be graphed by shading the region of the coordinate plane that satisfies the inequality. The boundary line of the shaded region is the graph of the linear equation Ax + B = C.

Linear inequalities can be used to solve a variety of problems, such as finding the maximum or minimum value of a function, or finding the range of values that a variable can take.

3. Solving Linear Inequalities, Unit 2 linear functions homework 7 linear inequalities

To solve a linear inequality, we can use the following steps:

1. Simplify the inequality by combining like terms and dividing or multiplying by the same non-zero number on both sides of the inequality.
2. Isolate the variable on one side of the inequality.
3. Check the solution by substituting it back into the original inequality.

There are a few different methods that can be used to solve linear inequalities. One method is to use the addition or subtraction property of inequality. This property states that if we add or subtract the same number from both sides of an inequality, the inequality remains true.

Another method that can be used to solve linear inequalities is to use the multiplication or division property of inequality. This property states that if we multiply or divide both sides of an inequality by the same positive number, the inequality remains true.

However, if we multiply or divide both sides of an inequality by the same negative number, the inequality is reversed.

4. Applications of Linear Inequalities

Linear inequalities are used in a variety of real-world applications. For example, linear inequalities can be used to:

• Find the maximum or minimum value of a function.
• Find the range of values that a variable can take.
• Solve problems involving optimization.
• Model real-world phenomena, such as the growth of a population or the cost of a product.

Linear inequalities are a powerful tool that can be used to solve a variety of problems. By understanding how to solve linear inequalities, we can better understand the world around us.

Commonly Asked Questions

What is the definition of a linear inequality?

A linear inequality is a mathematical statement that compares a linear expression to a constant using the inequality symbols ( <, >, ≤, ≥).

How do you graph a linear inequality?

To graph a linear inequality, first graph the corresponding linear equation as a solid line if the inequality is ≤ or ≥, or a dashed line if the inequality is < or >. Then, shade the region above the line if the inequality is > or ≥, or below the line if the inequality is< or ≤.

What are the different types of linear inequalities?

There are two main types of linear inequalities: one-variable inequalities and two-variable inequalities. One-variable inequalities involve a single variable, while two-variable inequalities involve two variables.