Question

Solve the inequality$x^2-3x<4$and graph the solution set.

Short answer

The solution set of inequality is $(-1,4)$and the graph is

Step-by-step solution

Step 1. Given Information

The given inequality is$x^2-3x<4$

Step 2. Calculation

Solve the inequality by rearranging it such that the right side has 0.

$$\begin{gathered} x^2-3x-4<4-4 \\ {x}^2-3x-4<0 \end{gathered}$$

Solve $f\left(0\right)$to find the y-intercept.

$$\begin{gathered} f\left(0\right)={\left(0\right)}^2-3\left(0\right)-4 \\ =-4 \end{gathered}$$

Factorize the equation.

$$\begin{gathered} (x-4)(x+1)=0 \\ x=4,-1 \end{gathered}$$

Thus, x-intercept are$4and-1$

Step 3. Calculation

On comparing the function with the general form, we get,

$a=1,b=-3,c=-4$

We will find the x- coordinate of the vertex as follows,

$$\begin{gathered} \frac{-b}{2a}=-\frac{-3}{2} \\ =\frac{3}{2} \end{gathered}$$

Solve $f\left(\frac{3}{2}\right)$to get the y-coordinate

$$\begin{gathered} f\left(\frac{3}{2}\right)={\left(\frac{3}{2}\right)}^2-3\left(\frac{3}{2}\right)-4 \\ =\frac{9}{4}-\frac{9}{2}-4 \\ =-\frac{25}{4} \end{gathered}$$

Graph the function by plotting all the points.

And the solution set is$\left\{x|-1<x<4\right\}$