Question

The figure shows a graphs of $y=5x-x^2$and$y=4$.

Use the graph to do the following.

(a) Find the solutions of the equation $5x-x^2=4$.

(b) Find the solutions of the inequality$5x-x^2>4$.

Short answer

(a) The real solution of the equation$5x-x^2=4$ are $x=1\text{and}x=4$.

(b) The solutions of the inequality$5x-x^2>4$ is the interval $\left(1,4\right)$.

Step-by-step solution

Part a. Step 1. Apply the concept of intercepts.

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on y-axis where the graph of the equation intersects the y-axis.

Part a. Step 2. Graph the equation.

The graph of the equations$y=5x-x^2$ and$y=4$ is provided below.

There solutions to equation are points of intersection of the curves.

Part a. Step 3. Find the intersection points.

Observe the points on the graph where the two curves intersect.

The points of intersection are $\left(1,4\right)\text{and}\left(4,4\right)$.

Thus, the real solution of the equation$5x-x^2=4$ are $x=1\text{and}x=4$.

Part b. Step 1. Interpret the inequality.

Consider the provided inequality $5x-x^2>4$.

Break the inequality in two equations.

Let the first equation be$y_1=5x-x^2$ and second equation be $y_2=4$.

Now, graph the equations on the same viewing window.

So, the inequality is rewritten as $y_1>y_2$.

Part b. Step 2. Graph the equations.

Consider the equations$y_1=5x-x^2$ and$y_2=4$

The graph of the equations$y_1=5x-x^2$ and$y_2=4$ is provided below.

Part b. Step 3. Find the solutions.

Observe the graph.

The point of intersection of the curves are $\left(1,4\right)\text{and}\left(4,4\right)$.

Part b. Step 4. Find the solution interval.

Consider the provided inequality $5x-x^2>4$.

Recall that the inequality is rewritten as$y_1>y_2$ wherewidth="83" height="27" role="math">y1=5x−x2and $y_2=4$.

So, the solution set contains all those x-coordinates for which $y_1>y_2$.

This means that all such x-coordinates where graph of the equation $y_1=5x-x^2$lies above the graph of the equation $y_2=4$.

Also, point of intersection of the graphs are $\left(1,4\right)\text{and}\left(4,4\right)$.

It is observed that the graph of the equation$y_1=5x-x^2$ lies above the graph of the equation$y_2=4$ when x coordinate is between 1 and 4, mathematically it is expressed as $x\in \left(1,4\right)$.

Thus, the solutions of the inequality$5x-x^2>4$ is the interval $\left(1,4\right)$.