Question

Which of the following ordered pairs is a solution to the graph of the system of inequalities? Select all that apply(5, 2)(-3, -4)(0, -3)(0, 1)(-4, 1)

Answer 1

(5, -2) and (0, -3)

Step-by-step explanation

We want to find which of the ordered pairs is a solution to the system of inequalities.

Ordered pairs are written in the form (x, y), this means, whichever ordered pair is a solution, when inserted into the system of inequalities, should be true.

This means that the values of x and y must be true for both inequalities in the system.

The system of inequalities is:

`\begin{cases}-2x-3\leq\text{ y} \\ y-1<\text{ }(1)/(2)x\end{cases}`

A. (5, -2)

`\begin{gathered} -2(5)\text{ - 3 }\leq-2\Rightarrow\text{ -10 - 3}\leq-2\Rightarrow\text{ -13 }\leq-2 \\ -2\text{ - 1 < }(1)/(2)(5)\Rightarrow\text{ -3 < }(5)/(2) \end{gathered}`

Since both inequalities are correct, this is a solution.

B. (-3, -4)

`-2(-3)\text{ - 3 }\leq-4\Rightarrow\text{ 6 - 3 }\leq-4\Rightarrow\text{ 3}\leq-4`

Since the first inequality is already incorrect, we do not need to go further.

It is not a solution

C. (0, -3)

`\begin{gathered} -2(0)\text{ - 3 }\leq\text{ -3 }\Rightarrow\text{ -3 }\leq\text{ -3} \\ -3\text{ - 1 < }(1)/(2)(0)\Rightarrow\text{ -4 < 0} \end{gathered}`

Since both inequalities are correct, this is a solution.

D. (0, 1)

`\begin{gathered} -2(0)\text{ - 3 }\leq\text{ 1 }\Rightarrow\text{ -3 }\leq\text{ 1} \\ 1\text{ - 1 < }(1)/(2)(0)\Rightarrow\text{ 0 < 0} \end{gathered}`

Since 0 is not less than 0, this is not a solution.

E. (-4, 1)

`-2(-4)\text{ - 3 }\leq\text{ 1}\Rightarrow\text{ 8 - 3 }\leq1\Rightarrow\text{ 5 }\leq1`

Since 5 is not less than 1, this is not a solution.

Therefore, the solutions are (5, -2) and (0, -3)