If $g(x)=|4-3x|+5$, Then $g(5) = \square$

Introduction

In mathematics, absolute value equations are a fundamental concept that can be used to solve various types of problems. The absolute value function is defined as the distance of a number from zero on the number line, without considering direction. In this article, we will focus on solving absolute value equations of the form $g(x) = |a - bx| + c$, where $a$, $b$, and $c$ are constants. We will use the given function $g(x) = |4 - 3x| + 5$ as an example to demonstrate the steps involved in solving absolute value equations.

Understanding Absolute Value Functions

An absolute value function is a function that takes the absolute value of an expression as its output. The absolute value of a number is its distance from zero on the number line, without considering direction. For example, the absolute value of $-3$ is $3$, and the absolute value of $3$ is also $3$. The absolute value function can be represented graphically as a V-shaped graph, with the vertex at the origin.

Solving Absolute Value Equations

To solve an absolute value equation of the form $g(x) = |a - bx| + c$, we need to follow these steps:

1. Isolate the absolute value expression: The first step is to isolate the absolute value expression on one side of the equation. In this case, we have $g(x) = |4 - 3x| + 5$, so we can isolate the absolute value expression by subtracting $5$ from both sides of the equation.
2. Set up two equations: Once we have isolated the absolute value expression, we can set up two equations by removing the absolute value sign and considering both the positive and negative cases.
3. Solve the two equations: We then solve the two equations separately to find the values of $x$ that satisfy the original equation.

Step-by-Step Solution

Let's apply these steps to the given function $g(x) = |4 - 3x| + 5$.

Step 1: Isolate the Absolute Value Expression

We can isolate the absolute value expression by subtracting $5$ from both sides of the equation:

$$g(x) - 5 = |4 - 3x|$$

Step 2: Set Up Two Equations

We can set up two equations by removing the absolute value sign and considering both the positive and negative cases:

$$4 - 3x = g(x) - 5 \quad \text{or} \quad 4 - 3x = -(g(x) - 5)$$

Step 3: Solve the Two Equations

We can solve the two equations separately to find the values of $x$ that satisfy the original equation.

Case 1: $4 - 3x = g(x) - 5$

We can substitute $g(x) = |4 - 3x| + 5$ into the equation:

$$4 - 3x = |4 - 3x| + 5 - 5$$

Simplifying the equation, we get:

$$4 - 3x = |4 - 3x|$$

We can remove the absolute value sign by considering both the positive and negative cases:

$$4 - 3x = 4 - 3x \quad \text{or} \quad 4 - 3x = -(4 - 3x)$$

Simplifying the equations, we get:

$$4 - 3x = 4 - 3x \quad \text{or} \quad 4 - 3x = -4 + 3x$$

The first equation is an identity, and the second equation can be simplified to:

$$8x = 8$$

Dividing both sides of the equation by $8$, we get:

$$x = 1$$

Case 2: $4 - 3x = -(g(x) - 5)$

We can substitute $g(x) = |4 - 3x| + 5$ into the equation:

$$4 - 3x = -|4 - 3x| - 5 + 5$$

Simplifying the equation, we get:

$$4 - 3x = -|4 - 3x|$$

We can remove the absolute value sign by considering both the positive and negative cases:

$$4 - 3x = -4 + 3x \quad \text{or} \quad 4 - 3x = 4 - 3x$$

The first equation can be simplified to:

$$8x = 8$$

Dividing both sides of the equation by $8$, we get:

$$x = 1$$

The second equation is an identity.

Conclusion

In this article, we have demonstrated the steps involved in solving absolute value equations of the form $g(x) = |a - bx| + c$. We have used the given function $g(x) = |4 - 3x| + 5$ as an example to illustrate the process. By following the steps outlined in this article, you can solve absolute value equations and gain a deeper understanding of this fundamental concept in mathematics.

Final Answer

The final answer to the problem is:

g(5) = |4 - 3(5)| + 5 = |-11| + 5 = 16$<br/> **Absolute Value Equations: A Q&A Guide** ===================================== **Introduction** --------------- In our previous article, we discussed the steps involved in solving absolute value equations of the form $g(x) = |a - bx| + c$. In this article, we will provide a Q&A guide to help you better understand the concept of absolute value equations and how to solve them. **Q: What is an absolute value equation?** ----------------------------------------- A: An absolute value equation is a type of equation that involves the absolute value of an expression. The absolute value of a number is its distance from zero on the number line, without considering direction. **Q: How do I solve an absolute value equation?** ---------------------------------------------- A: To solve an absolute value equation, you need to follow these steps: 1. **Isolate the absolute value expression**: The first step is to isolate the absolute value expression on one side of the equation. 2. **Set up two equations**: Once you have isolated the absolute value expression, you can set up two equations by removing the absolute value sign and considering both the positive and negative cases. 3. **Solve the two equations**: You then solve the two equations separately to find the values of $x$ that satisfy the original equation. **Q: What is the difference between an absolute value equation and a linear equation?** -------------------------------------------------------------------------------- A: An absolute value equation is a type of equation that involves the absolute value of an expression, while a linear equation is a type of equation that involves a linear expression. For example, the equation $|x| = 3$ is an absolute value equation, while the equation $x = 3$ is a linear equation. **Q: Can I use the same steps to solve absolute value inequalities?** ---------------------------------------------------------------- A: No, you cannot use the same steps to solve absolute value inequalities. Absolute value inequalities involve the absolute value of an expression, but they also involve a comparison to a value. To solve absolute value inequalities, you need to follow a different set of steps. **Q: How do I know which case to use when solving an absolute value equation?** ------------------------------------------------------------------------- A: When solving an absolute value equation, you need to consider both the positive and negative cases. To determine which case to use, you need to look at the expression inside the absolute value sign and determine whether it is positive or negative. **Q: Can I use absolute value equations to model real-world problems?** ---------------------------------------------------------------- A: Yes, you can use absolute value equations to model real-world problems. For example, you can use absolute value equations to model the distance between two points, the cost of a product, or the temperature of a system. **Q: What are some common applications of absolute value equations?** ---------------------------------------------------------------- A: Some common applications of absolute value equations include: * **Distance problems**: Absolute value equations can be used to model the distance between two points. * **Cost problems**: Absolute value equations can be used to model the cost of a product. * **Temperature problems**: Absolute value equations can be used to model the temperature of a system. * **Finance problems**: Absolute value equations can be used to model the value of a stock or a bond. **Q: How do I graph an absolute value equation?** ------------------------------------------------ A: To graph an absolute value equation, you need to follow these steps: 1. **Plot the vertex**: The vertex of the graph is the point where the absolute value expression is equal to zero. 2. **Plot the two arms**: The two arms of the graph are the two lines that extend from the vertex. 3. **Determine the direction of the arms**: The direction of the arms depends on the sign of the coefficient of the absolute value expression. **Conclusion** ---------- In this article, we have provided a Q&A guide to help you better understand the concept of absolute value equations and how to solve them. We have also discussed some common applications of absolute value equations and how to graph them. By following the steps outlined in this article, you can gain a deeper understanding of absolute value equations and how to use them to model real-world problems. **Final Tips** -------------- * **Practice, practice, practice**: The best way to learn how to solve absolute value equations is to practice, practice, practice. * **Use visual aids**: Visual aids such as graphs and charts can help you to better understand the concept of absolute value equations. * **Break down complex problems**: Complex problems can be broken down into simpler sub-problems that can be solved using absolute value equations.