/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 78 Let \(f(x)=\frac{3}{2} x-2\) For... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 78

Let \(f(x)=\frac{3}{2} x-2\) For what value of \(x\) does function \(f\) have the given value? \(f(x)=\frac{2}{3}\)

Short Answer

Expert verified
The value of \(x\) is \(\frac{16}{9}\).

Step by step solution

01

Set the Function Equal to the Given Value

To find the value of \(x\) for which the function \(f(x)\) equals \(\frac{2}{3}\), we start by setting the equation \(\frac{3}{2} x - 2 = \frac{2}{3}\).
02

Add 2 to Both Sides

Next, to isolate the term with \(x\), add 2 to both sides of the equation: \[\frac{3}{2} x - 2 + 2 = \frac{2}{3} + 2\]This simplifies to: \[\frac{3}{2} x = \frac{2}{3} + 2\]
03

Convert 2 to a Fraction

Before adding, convert the integer \(2\) into a fraction with common denominator 3: \[2 = \frac{6}{3}\]Now substitute back into the equation:\[\frac{3}{2} x = \frac{2}{3} + \frac{6}{3}\]
04

Add the Fractions

Add the two fractions on the right side of the equation: \[\frac{2}{3} + \frac{6}{3} = \frac{8}{3}\]Now the equation is:\[\frac{3}{2} x = \frac{8}{3}\]
05

Solve for x

To solve for \(x\), multiply both sides by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\): \[x = \frac{8}{3} \times \frac{2}{3}\]Calculate the right side:\[x = \frac{16}{9}\]

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Key Concepts

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

Linear Function
A linear function is a type of function where the graph is a straight line. It has the general form \(f(x) = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. In our exercise, the function \(f(x) = \frac{3}{2} x - 2\) is a linear function, where \(\frac{3}{2}\) is the slope and \(-2\) is the y-intercept.

  • **Slope \(m\)**: This indicates the steepness and the direction of the line. A positive slope means the line rises as it moves from left to right.
  • **Y-intercept \(b\)**: It is the point where the line crosses the y-axis, meaning when \(x = 0\).
Linear functions are widely used because they describe proportional relationships and are simple to calculate and interpret. Recognizing a linear function helps in predicting and analyzing situations that vary at a constant rate.
Solving Equations
One key skill in mathematics is solving equations, which is a process to determine the values that satisfy a given equation. To solve \(\frac{3}{2} x - 2 = \frac{2}{3}\), we must find the value of \(x\) that makes the equation true. The process involves several fundamental steps:

  • **Isolate the variable**: Start by moving constant terms to one side of the equation to focus on the term containing the variable \(x\).
  • **Modify terms**: When equations involve fractions, convert integers to fractions to simplify the process of combining terms.
  • **Solve by inverse operations**: Once the variable \(x\) is isolated, use inverse operations such as division to solve for \(x\). Here, multiplying by the reciprocal helps to clear fractions, providing the solution \(x = \frac{16}{9}\).
Understanding how to systematically approach an equation fosters confidence in navigating both simple and complex equations.
Fractions
Fractions are an essential concept in mathematics, representing a part of a whole and playing a critical role in solving equations. A fraction consists of a numerator and a denominator, denoted as \(\frac{a}{b}\).

When solving problems involving fractions:
  • **Conversion**: Convert whole numbers to fractions to facilitate addition or subtraction with existing fractions. This involves writing the number as a fraction with the desired denominator, like converting 2 to \(\frac{6}{3}\).
  • **Addition and Subtraction**: To add or subtract fractions, first ensure they have common denominators. Only then can you combine the numerators while keeping the denominator the same.
  • **Multiplication**: Multiply numerators and denominators straight across. For example, multiplying fractions \(\frac{8}{3}\) and \(\frac{2}{3}\) yields \(\frac{16}{9}\).
Mastering fractions improves one's ability to solve equations that include them, as seen in this exercise, and aids in interpreting mathematical results.

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Most popular questions from this chapter

a. Suppose you know the slope of a line. Is that enough information about the line to write its equation? Explain. b. Suppose you know the coordinates of a point on a line. Is that enough information about the line to write its equation? Explain.

Solve each formula for the indicated variable. $$ T-W=m a \text { for } W $$

Windchill. A combination of cold and wind makes a person feel colder than the actual temperature. The table shows what temperatures of \(35^{\circ} \mathrm{F}\) and \(15^{\circ} \mathrm{F}\) feel like when a 15 -mph wind is blowing. The relationship between the actual temperature and the windchill temperature can be modeled with a linear equation. a. Write the equation that models this relationship. Answer in slope-intercept form. b. What information is given by the \(y\) -intercept of the graph of the equation found in part (a)? $$ \begin{array}{|c|c|} \hline \multicolumn{2}{|c|}\text { 15-mph wind } \\ \hline \text { Actual temperature } \boldsymbol{x} & \text { Windchill temperature } \boldsymbol{y} \\ \hline 35^{\circ} \mathrm{F} & 25^{\circ} \mathrm{F} \\ \hline 15^{\circ} \mathrm{F} & 0^{\circ} \mathrm{F} \\ \hline \end{array} $$

Write an equation for a linear function whose graph has the given characteristics. Passes through ( \(3,0\) ), parallel to the graph of \(g(x)=-\frac{2}{3} x-4\)

Graph function. \(f(x)=-4\)
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