Question

Solving Equation and Inequalities Graphically Solve the given equation or inequality graphically.

(a)$-x^2=3-4x$

(b)$-x^2\ge 3-4x$

Short answer

(a) The solution of the equation$-x^2=3-4x$ is $x=1,3$.

(b) The solution of the inequality$-x^2\ge 3-4x$ is $x\le 1\text{\hspace{0.33em}or}x\ge 3$.

Step-by-step solution

Part a. Step 1. Write down the given equation.

The given equation is

$-x^2=3-4x\mathrm{...}\left(1\right)$

In equation (1) functions f and g corresponds to the left hand side and right hand side of the equations respectively. So,

$\begin{array}{l}f\left(x\right)=-x^2\\ g\left(x\right)=3-4x\end{array}$

Part a. Step 2. Draw the graph.

The graph of g and f on the same set of axis is shown below.

Figure (1)

Part a. Step 3. Find the solution of the equation from graph.

The given function is equivalent to$f\left(x\right)=g\left(x\right)$ and from the graph of these function the solutions of the equation is$1$ and$3$ because at$x=1,3$ the functions intersect with each other.

Thus, the solution of the equation$-x^2=3-4x$ is $x=1,3$.

Part b. Step 1. Write down the given inequality.

The given inequality is

$-x^2\ge 3-4x\mathrm{...}\left(2\right)$

In equation (1) functions f and g corresponds to the left hand side and right hand side of the equations respectively. So,

$\begin{array}{l}f\left(x\right)=-x^2\\ g\left(x\right)=3-4x\end{array}$

Part b. Step 2. Draw the graph.

The graph of g and f on the same set of axis is shown below.

Figure (2)

Part b. Step 3. Find the solution of the inequality from graph.

The given function is equivalent to$f\left(x\right)\ge g\left(x\right)$ and from the graph of these function the solutions of the equation is$\left(-\infty ,1\right]\text{or}\left[3,\infty \right)$ because the function$f\left(x\right)$ is greater than and equal to the function$g\left(x\right)$ when$x\le 1$ or $x\ge 3$.

Thus, the solution of the inequality$-x^2\ge 3-4x$ is $x\le 1\text{\hspace{0.33em}or}x\ge 3$.