/*! 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 100 If \(f(x)=\frac{5}{6} x-\frac{3}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 100

If \(f(x)=\frac{5}{6} x-\frac{3}{4},\) find the value \((s)\) of \(x\) so that \(f(x)=-\frac{7}{16}\)

Short Answer

Expert verified
The value of \(x\) is \( \frac{3}{8} \).

Step by step solution

01

- Set Up the Equation

Given the function \( f(x) = \frac{5}{6}x - \frac{3}{4} \) and the condition \( f(x) = -\frac{7}{16} \), we set up the equation: \( \frac{5}{6}x - \frac{3}{4} = -\frac{7}{16} \).
02

- Isolate the Linear Term

To isolate the term involving \(x\), add \( \frac{3}{4} \) to both sides of the equation: \( \frac{5}{6}x = -\frac{7}{16} + \frac{3}{4} \).
03

- Find a Common Denominator

Convert \( \frac{3}{4} \) to a fraction with a denominator of 16: \( \frac{3}{4} = \frac{12}{16} \). The equation now reads: \( \frac{5}{6}x = -\frac{7}{16} + \frac{12}{16} \).
04

- Simplify the Right Side

Combine the fractions on the right side: \( \frac{5}{6}x = \frac{12}{16} - \frac{7}{16} = \frac{5}{16} \).
05

- Solve for x

To solve for \(x\), multiply both sides by the reciprocal of \( \frac{5}{6} \), which is \( \frac{6}{5} \): \( x = \frac{5}{16} \times \frac{6}{5} = \frac{6}{16} = \frac{3}{8} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Solving Linear Equations
Solving linear equations is a foundational skill in algebra. A linear equation typically has the form: y = mx + bwhere m and b are constants. To solve a linear equation, follow these steps:
  • Isolate the variable (x): Your goal is to get x by itself on one side of the equation. You do this through operations like addition, subtraction, multiplication, and division.
  • Combine like terms and simplify: If the equation has terms that can be combined, such as constant numbers, do so to simplify the equation.
  • Check your solution: Substitute your solution back into the original equation to ensure it works.
Simplifying and isolating x might involve additional small steps to handle fractions or rearrange terms, but the core idea is always to get x alone.
Function Notation
Function notation is a shorthand way of representing functions. It uses the format f(x) to denote a function named f with x as the input variable. For example, f(x) = mx + b can describe a linear function, where m represents the slope and b the y-intercept. Here are a few points to note:
  • Evaluating functions: To find f(a) for some value a, replace x with a in the function鈥檚 formula.
  • Equality in functions: Setting f(x) equal to a value lets you solve for x, as in the exercise above where f(x) = -7/16.
  • Visual interpretations: Functions can be graphed on a coordinate plane, showing how y changes with x.
Understanding function notation makes it easier to work with and solve various types of algebraic problems.
Fractions in Equations
Handling fractions in equations can seem tricky, but there are strategies to make it easier:
  • Common denominators: When adding or subtracting fractions, convert them to have a common denominator. For example, in the exercise, \[ \frac{3}{4} \] became \[ \frac{12}{16} \] to match \[ \frac{7}{16} \]. This simplifies combining the fractions.
  • Reciprocals: To clear fractions involving the variable, use the reciprocal. In the exercise, multiplying both sides by \[ \frac{6}{5} \]removed the fractional coefficient of x.
  • Avoiding Errors: Be careful with signs and arithmetic operations. Double-check each step.
Mastering these concepts ensures fractions won鈥檛 be a stumbling block in linear equations.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to isolate variables or simplify expressions. Key techniques include:
  • Combining like terms: Simplify the equation by merging terms with the same variable or constant.
  • Using properties of equality: Add, subtract, multiply, or divide both sides of the equation by the same number to maintain equality.
  • Simplifying expressions: Break down complex expressions into simpler components to make manipulation easier, like converting \[ \frac{3}{4} \] to \[ \frac{12}{16} \].
These skills are crucial not just in linear equations but across diverse topics in mathematics. They provide a toolkit for tackling increasingly complex problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Stopping Distance When the driver of a vehicle observes an impediment, the total stopping distance involves both the reaction distance \(R\) (the distance the vehicle travels while the driver moves his or her foot to the brake pedal) and the braking distance \(B\) (the distance the vehicle travels once the brakes are applied). For a car traveling at a speed of \(v\) miles per hour, the reaction distance \(R\), in feet, can be estimated by \(R(v)=2.2 v .\) Suppose that the braking distance \(B,\) in feet, for a car is given by \(B(v)=0.05 v^{2}+0.4 v-15\) (a) Find the stopping distance function $$ D(v)=R(v)+B(v) $$ (b) Find the stopping distance if the car is traveling at a speed of \(60 \mathrm{mph}\). (c) Interpret \(D(60)\)

Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: (0,-3) and (3,0)\(;\) a local maximum value of -2 at \(-1 ;\) a local minimum value of -6 at \(2 .\) Compare your graph with those of others. Comment on any differences.

The daily rental charge for a moving truck is \(\$ 40\) plus a mileage charge of \(\$ 0.80\) per mile. Express the cost \(C\) to rent a moving truck for one day as a function of the number \(x\) of miles driven.

The period \(T\) (in seconds) of a simple pendulum is a function of its length \(l\) (in feet) defined by the equation $$ T=2 \pi \sqrt{\frac{l}{g}} $$ where \(g \approx 32.2\) feet per second per second is the acceleration due to gravity. (a) Use a graphing utility to graph the function \(T=T(l)\). (b) Now graph the functions \(T=T(l+1), T=T(l+2)\) $$ \text { and } T=T(l+3) $$ (c) Discuss how adding to the length \(l\) changes the period \(T\) (d) Now graph the functions \(T=T(2 l), T=T(3 l)\), and \(T=T(4 l)\) (e) Discuss how multiplying the length \(l\) by factors of 2,3 , and 4 changes the period \(T\)

Motion of a Golf Ball A golf ball is hit with an initial velocity of 130 feet per second at an inclination of \(45^{\circ}\) to the horizontal. In physics, it is established that the height \(h\) of the golf ball is given by the function $$ h(x)=\frac{-32 x^{2}}{130^{2}}+x $$ where \(x\) is the horizontal distance that the golf ball has traveled. (a) Determine the height of the golf ball after it has traveled 100 feet. (b) What is the height after it has traveled 300 feet? (c) What is \(h(500) ?\) Interpret this value. (d) How far was the golf ball hit? (e) Use a graphing utility to graph the function \(h=h(x)\). (f) Use a graphing utility to determine the distance that the ball has traveled when the height of the ball is 90 feet. (g) Create a TABLE with TblStart \(=0\) and \(\Delta \mathrm{Tbl}=25 .\) To the nearest 25 feet, how far does the ball travel before it reaches a maximum height? What is the maximum height? (h) Adjust the value of \(\Delta\) Tbl until you determine the distance, to within 1 foot, that the ball travels before it reaches its maximum height.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.