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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.
\[ \left\{ \begin{array}{l} \text{first sentence} \\ \text{second sentence} \end{array} \right. \]
(Type inequalities. Simplify your answer.)
Solution
The problem requires writing a system of inequalities from given sentences and graphing it. Let’s analyze the sentences and write the inequalities: 1. “The sum of the x-variable and the y-variable is no more than 3.” This can be written as: \[ x + y \leq 3 \] 2. “The y-variable is no less than the difference between the square of the x-variable and 4.” This can be written as: \[ y \geq x^2 - 4 \] Thus, the system of inequalities is: \[ \begin{cases} x + y \leq 3 & \\ y \geq x^2 - 4 & \end{cases} \]