Question: Write the given sentences as a system of inequalities in two variables. Then graph the system. The…

Write the given sentences as a system of inequalities in two variables. Then graph the system.

The sum of the x-variable and the y-variable is no more than 3. The y-variable is no less than the difference between the square of the x-variable and 4.

Complete the system of inequalities in two variables.

\[ \begin{cases} x + y \leq 3 & \text{first sentence} \\ y \geq x^2 - 4 & \text{second sentence} \end{cases} \]

(Type inequalities. Simplify your answer.)

Use the graphing tool to graph the inequalities.

Click to enlarge graph

Solution

The problem provides a verbal description of a system of inequalities and asks to complete and graph it: 1. The sum of the x-variable and the y-variable is no more than 3: \[ x + y \leq 3 \] 2. The y-variable is no less than the difference between the square of the x-variable and 4: \[ y \geq x^2 - 4 \] The complete system of inequalities is: \[ \begin{cases} x + y \leq 3 & \\ y \geq x^2 - 4 & \end{cases} \]