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Chapter 4: Problem 19

$$f(x)=-3 x-2 . \text { Find } x \text { so that } f(x)=10$$

Short Answer

Expert verified
The value of \(x\) that makes \(f(x)=-3x-2\) equal to 10 is \(x=-4\).

Step by step solution

01

Substitute f(x) with 10

To find the value of \(x\) for which \(f(x) = 10\), we substitute \(f(x)\) with \(10\). This gives us the equation: \[10 = -3x - 2\]
02

Solve for x

Now, we need to solve the equation for \(x\). We can do this by isolating \(x\) as follows: Add 2 to both sides of the equation: \[10 + 2 = (-3x - 2) + 2\] This simplifies to: \[12 = -3x\] Now, we can solve for \(x\) by dividing both sides of the equation by -3: \[\frac{12}{-3} = \frac{-3x}{-3}\] This gives us the solution: \[x = -4\] So, when \(x = -4\), the function \(f(x) = -3x - 2\) is equal to 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving for x
When dealing with equations, "solving for x" means finding the value of the variable \( x \) that satisfies the equation. Consider the linear equation we derived:
  • 10 = -3x - 2
The goal is to isolate \( x \) on one side of the equation, allowing us to determine its exact value. To do this, follow these steps:
  • Add 2 to both sides of the equation to eliminate the constant term from the right side. This transforms the initial equation to:
  • 12 = -3x
  • Next, divide both sides by -3, the coefficient of \( x \), to solve for \( x \)
  • This operation simplifies to \( x = -4 \).
These steps outline the basic approach to solving for \( x \), a crucial aspect of understanding and working with linear equations.
Function Notation
Function notation is a way of expressing relationships between variables. In our example, the function is denoted as \( f(x) = -3x - 2 \). This notation communicates several important things:
  • The expression, \(-3x - 2\), represents a function of \( x \).
  • "\( f(x) \)" indicates the output of function \( f \) when the input is \( x \).
In the problem, we set \( f(x) = 10 \). This tells us that we need to find the input \( x \) such that the output of the function equals 10. Function notation is a compact and powerful way to define and work with mathematical relationships, especially when dealing with equations and graphs. It helps in easily substituting values and understanding the functional relationship between variables.
Algebraic Manipulation
Algebraic manipulation involves reorganizing and simplifying equations to solve for a variable, like \( x \) in our example. This skill is essential for transforming complex expressions into simpler, more manageable forms. To effectively perform algebraic manipulation, consider these methods:
  • Use the properties of equality, such as adding, subtracting, multiplying, or dividing both sides of an equation by the same number to maintain balance.
  • Reorder terms to make the equation more straightforward, often by getting all terms involving the variable on one side and constants on the other.
  • Simplify expressions by combining like terms or using distributive properties if applicable.
By mastering algebraic manipulation, solving equations becomes logical and systematic, allowing for clear paths to find solutions. This skill is fundamental in solving a wide range of mathematical problems beyond basic equations, making it a vital tool in any algebraic toolbox.

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Most popular questions from this chapter

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=-1\) and contains \((8,-8)\)

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