Question

Determine whether each ordered pair is a solution to the inequality $y<x+1$:

$$\begin{gathered} \mathit{a}\mathbf{)}(0,0\left) \\ \mathit{b}\mathbf{\right)}(8,6) \\ \mathit{c}\mathbf{)}(-2,-1\left) \\ \mathit{d}\mathbf{\right)}(3,4) \\ \mathit{e}\mathbf{)}(-1,-4) \end{gathered}$$

Short answer

The ordered pairs which are solutions to the inequality $y<x+1$are $\mathit{a}\mathbf{)}(0,0)\mathit{b}\mathbf{)}(8,6)\mathit{e}\mathbf{)}(-1,-4)$.

The ordered pairs $\mathit{c}\mathbf{)}(-2,-1)\mathit{d}\mathbf{)}(3,4)$are not a solution to the given inequality.

Step-by-step solution

Step 1. Given information

We are given inequality $y<x+1$and the ordered pairs as follows:

$$\begin{gathered} \mathit{a}\mathbf{)}\mathbf{}(0,0\left) \\ \mathit{b}\mathbf{\right)}\mathbf{}(8,6) \\ \mathit{c}\mathbf{)}\mathbf{}(-2,-1\left) \\ \mathit{d}\mathbf{\right)}(3,4) \\ \mathit{e}\mathbf{)}(-1,-4) \end{gathered}$$

We have to determine which of the above ordered pairs is a solution to the given inequality.

Note that, an ordered pair $(x,y)$is a solution to a linear inequality if that inequality is true when we substitute the values for$x$and$y$.

Step 2. Check the ordered pair a) (0,0)

The given ordered pair is $(0,0)$

Put $x=0,y=0$in the given inequality; we have

$$\begin{gathered} y<x+1 \\ 0<0+1 \\ 0<1 \end{gathered}$$

Which is true.

Therefore the ordered pair$(0,0)$is a solution to the given inequality.

Step 3. Check the ordered pair b) (8,6)

The given ordered pair is $(8,6)$.

Substituting the values $x=8,y=6$in the inequality $y<x+1$, we have

$$\begin{gathered} y<x+1 \\ 6<8+1 \\ 6<9 \end{gathered}$$

which is true.

Therefore, the ordered pair$(8,6)$is a solution to the given inequality.

Step 4. Check the ordered pair c) (-2,-1)

The given ordered pair is $(-2,-1)$

Substitute $x=-2,y=-1$in the inequality $y<x+1$, we have role="math" localid="1644903129231" $$\begin{gathered} y<x+1 \\ -1<-2+1 \\ -1<-1 \end{gathered}$$

which is not true.

Therefore, the ordered pair$(-2,-1)$is not a solution to the given inequality.

Step 5. Check for the ordered pair d) (3,4)

The given ordered pair is $(3,4)$

Substituting $x=3,y=4$in the given inequality, we have

$$\begin{gathered} y<x+1 \\ 4<3+1 \\ 4<4 \end{gathered}$$

which is not true.

Therefore, the ordered pair$(3,4)$is not a solution to the given inequality.

Step 6. Check the ordered pair e) (-1,-4)

The given ordered pair is $(-1,-4)$

Substituting $x=-1,y=-4$in the inequality$y<x+1$, we have

$$\begin{gathered} y<x+1 \\ -4<-1+1 \\ -4<0 \end{gathered}$$

which is true.

Thus the ordered pair$(-1,-4)$is a solution to the inequality$y<x+1$.

Step 7. Final answer

The ordered pairs that aresolutions to the inequality $y<x+1$are,

$$\begin{gathered} \mathit{a}\mathbf{)}(0,0\left) \\ \mathit{b}\mathbf{\right)}(8,6) \\ \mathit{e}\mathbf{)}(-1,-4) \end{gathered}$$

The ordered pairs that are not solutions to the given inequality are,

$$\begin{gathered} \mathit{c}\mathbf{)}(-2,-1\left) \\ \mathit{d}\mathbf{\right)}(3,4) \end{gathered}$$