Question

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

(a)$-2x+3=3x-7$

(b)$-2x+3\le 3x-7$

Short answer

(a) The solution of the equation$-2x+3=3x-7$ is $x=2$.

(b) The solution of the inequality$-2x+3\le 3x-7$ is $x\ge 2$.

Step-by-step solution

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

The given equation is

$-2x+3=3x-7\mathrm{...}\left(1\right)$

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

$\begin{array}{l}f\left(x\right)=-2x+3\\ g\left(x\right)=3x-7\end{array}$

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$2$ because at$x=2$ the functions intersect with each other.

Thus, the solution of the equation$-2x+3=3x-7$ is $x=2$.

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

The given inequality is

$-2x+3\le 3x-7\mathrm{...}\left(2\right)$

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

$\begin{array}{l}f\left(x\right)=-2x+3\\ g\left(x\right)=3x-7\end{array}$

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$f\left(x\right)\le g\left(x\right)$ and from the graph of these function the solutions of the equation is$\left(2,\infty \right)$ because the function$f\left(x\right)$ is less than or equal to the function$g\left(x\right)$ when $x\ge 2$.

Thus, the solution of the inequality$-2x+3\le 3x-7$ is $x\ge 2$.