Question

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

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

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 \(x = \frac{1}{4}\).

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 \(x > \frac{1}{4}\).

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 \(x = 1\).

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 \(x \leq 1\).

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 \(f(x) = g(x)\).

Key concepts

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.