/*! 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 29 Suppose that \(f(x)=4 x-1\) and ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 29

Suppose that \(f(x)=4 x-1\) and \(g(x)=-2 x+5\) (a) Solve \(f(x)=0\) (b) Solve \(f(x)>0\) (c) Solve \(f(x)=g(x)\) (d) Solve \(f(x) \leq g(x)\). (e) Graph \(y=f(x)\) and \(y=g(x)\) and label the point that represents the solution to the equation \(f(x)=g(x)\).

Short Answer

Expert verified
Part (a): \(x = \frac{1}{4}\). Part (b): \(x > \frac{1}{4}\). Part (c): \(x = 1\). Part (d): \(x \leq 1\). Part (e): The intersection point is \((1, 3)\).

Step by step solution

01

Solve Part (a) - Solve for \(f(x) = 0\)

Start by setting the function \(f(x) = 4x - 1\) equal to zero: \[ 4x - 1 = 0 \] Add 1 to both sides: \[ 4x = 1 \] Divide both sides by 4: \[ x = \frac{1}{4} \] Therefore, the solution is \(x = \frac{1}{4}\).
02

Solve Part (b) - Solve for \(f(x) > 0\)

Set the function \(f(x) = 4x - 1\) greater than zero: \[ 4x - 1 > 0 \] Add 1 to both sides: \[ 4x > 1 \] Divide both sides by 4: \[ x > \frac{1}{4} \] Therefore, the solution is \(x > \frac{1}{4}\).
03

Solve Part (c) - Solve for \(f(x) = g(x)\)

Set the functions \(f(x) = 4x - 1\) and \(g(x) = -2x + 5\) equal to each other: \[ 4x - 1 = -2x + 5 \] Add \(2x\) to both sides: \[ 6x - 1 = 5 \] Add 1 to both sides: \[ 6x = 6 \] Divide both sides by 6: \[ x = 1 \] Therefore, the solution is \(x = 1\).
04

Solve Part (d) - Solve for \(f(x) \leq g(x)\)

Set the functions \(f(x) = 4x - 1\) and \(g(x) = -2x + 5\) such that \(f(x) \leq g(x)\): \[ 4x - 1 \leq -2x + 5 \] Add \(2x\) to both sides: \[ 6x - 1 \leq 5 \] Add 1 to both sides: \[ 6x \leq 6 \] Divide both sides by 6: \[ x \leq 1 \] Therefore, the solution is \(x \leq 1\).
05

Graph Part (e) - Graph \(y=f(x)\) and \(y=g(x)\)

Graph the linear functions \(y = 4x - 1\) and \(y = -2x + 5\). These are both straight lines. The point of intersection is where \(x = 1\) from part (c). For \(x = 1\): \[ y = 4(1) - 1 = 3 \] Thus, the point of intersection is \((1, 3)\). Label the point \((1, 3)\) on the graph as the solution to the equation \(f(x) = g(x)\).

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Key Concepts

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

solving linear equations
Solving linear equations involves finding the value of the variable that makes the equation true. A linear equation is an equation of the form $$ax + b = c$$.For example, to solve the equation $$4x - 1 = 0$$:
  • First, add 1 to both sides: $$4x = 1$$
  • Then, divide both sides by 4 to isolate x: $$x = \frac{1}{4}$$
This means the solution is $$x = \frac{1}{4}$$. Linear equations are fundamental in algebra because they appear frequently in different contexts and are the foundation for more complex equations.
inequalities
Inequalities express a relationship where one side is not strictly equal to the other but is either greater than, less than, or equal to the other side. For example, let's solve the inequality $$4x - 1 > 0$$:
  • Add 1 to both sides: $$4x > 1$$
  • Divide both sides by 4: $$x > \frac{1}{4}$$
This shows that $$x$$ must be greater than $$\frac{1}{4}$$. Inequalities are used to express ranges of possible solutions and can be graphed on a number line, highlighting the range of values that satisfy the inequality.
graphing linear functions
Graphing linear functions involves plotting points on a coordinate plane that form a straight line. A linear function can be written as $$y = mx + b$$, where $$m$$ is the slope, and $$b$$ is the y-intercept. For example, let's graph the function $$y = 4x - 1$$:
  • Find the y-intercept ($$b=-1$$), which is where the line crosses the y-axis.
  • Use the slope ($$m=4$$) to determine the rise over run from the y-intercept.
To plot a second function like $$y = -2x + 5$$, follow the same steps. The graph helps visualize the behavior of the functions and the points where they intersect.
intersection of lines
The intersection of lines on a graph represents the point at which they intersect or cross each other, providing the solution to both equations simultaneously. For instance, solving the system of equations $$4x - 1 = -2x + 5$$ helps find the intersection:
  • Merge terms:$$6x - 1 = 5$$
  • Add 1 to both sides:$$6x = 6$$
  • Divide by 6:$$x = 1$$
Now, substitute $$x = 1$$ back into either equation to find the y-coordinate:
  • For $$y = 4(1) - 1$$,$$y = 3$$
So, the intersection point is (1, 3). Graphing this on the coordinate plane as the point where the two lines meet visually confirms the solution.

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Most popular questions from this chapter

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A self-catalytic chemical reaction results in the formation of a compound that causes the formation ratio to increase. If the reaction rate \(V\) is modeled by $$ V(x)=k x(a-x), \quad 0 \leq x \leq a $$ where \(k\) is a positive constant, \(a\) is the initial amount of the compound, and \(x\) is the variable amount of the compound, for what value of \(x\) is the reaction rate a maximum?
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