Question

Which of the following is a solution to the following system of inequalities?

solid line at y equals negative x plus one, dotted line at y equals two x plus 3, shaded above the first line and below the second line

(1, -1)

(-4, 0)

(3, -2)

(0, 3)

Answer 1

Option D:
`(0,3)` is the solution to the inequalities.

Step-by-step explanation:

From the given graph, we can see that the equation of the inequalities are

`$y>-x+1$` and
`$y\leq 2 x+3$`

To determine the coordinate that satisfies the inequality, let us substitute the coordinates in both of the inequalities
`$y>-x+1$` and
`$y<2 x+3$`

Thus, we have,

Option A:
`(1,-1)`

Substituting the coordinates in
`$y>-x+1$` and
`$y\leq 2 x+3$`, we get,

`y>-x+1\implies-1>0` is not true.

`$y\leq 2 x+3 \implies -1\leq 5` is true.

Since, only one equation satisfies the condition, the coordinate
`(1,-1)` is not a solution.

Hence, Option A is not the correct answer.

Option B:
`(-4,0)`

Substituting the coordinates in
`$y>-x+1$` and
`$y\leq 2 x+3$`, we get,

`y>-x+1\implies0>5` is not true.

`$y\leq 2 x+3 \implies 0\leq -5` is not true.

Since, both the equations does not satisfy the condition, the coordinate
`(-4,0)` is not a solution.

Hence, Option B is not the correct answer.

Option C:
`(3,-2)`

Substituting the coordinates in
`$y>-x+1$` and
`$y\leq 2 x+3$`, we get,

`y>-x+1\implies-2>-2` is not true.

`$y\leq 2 x+3 \implies -2\leq 9` is true.

Since, only one equation satisfies the condition, the coordinate
`(3,-2)` is not a solution.

Hence, Option C is not the correct answer.

Option D:
`(0,3)`

Substituting the coordinates in
`$y>-x+1$` and
`$y\leq 2 x+3$`, we get,

`y>-x+1\implies3>1` is true.

`$y\leq 2 x+3 \implies 3\leq 3` is true.

Since, both equation satisfies the condition, the coordinate
`(0,3)` is a solution.

Hence, Option D is the correct answer.