Question

Determine which point is part of the solution set to the followingsystem of inequalities: f(x) = 2x + 2; f(x) = -4x + 3; andf(x) s 6x+ 5.(0,0)(-2.5, 0)(0, 6.5)(0,2.5)

Answer 1

The point that is part of the set of solutions is (0,2.5)

To solve this, we need to try each point in each equation to find the one that verifies the three equations.

Let's see the first one.

`f\mleft(x\mright)\ge2x+2`

replacing 0 in the x and 0 in the f(x):

`\begin{gathered} 0\ge2\cdot0+2 \\ 0\ge2 \end{gathered}`

Wich is false, so we can rule out the first point.

For the second point, (-2.5,0):

`\begin{gathered} 0\ge2(-2.5)+2=-5+2=-3 \\ 0\ge-3 \end{gathered}`

So is solution of the first equation. Let's see the second equation.

`\begin{gathered} f\mleft(x\mright)\leq-4x+3\text{ }\Rightarrow0\leq-4(-2.5)+3=10+3 \\ 0\leq13 \end{gathered}`

Also true, Now the third equation

`\begin{gathered} f\mleft(x\mright)\leq6x+5\Rightarrow0\leq6(-2.5)+5=-10 \\ 0\leq-10 \end{gathered}`

This is false, so we can rule out the second point.

For the third point (0,6.5). The first equation:

`\begin{gathered} f(x)\ge2x+2\Rightarrow6.5\ge2\cdot0+2 \\ 6.5\ge2 \end{gathered}`

Is true. Second equation

`\begin{gathered} f(x)\leq-4x+3\Rightarrow6.5\leq-4\cdot0+3 \\ 6.5\leq3 \end{gathered}`

Is false.

We have ruled out all points except the last one. Lets see that is a solution for all equations

`\begin{gathered} f(x)\ge2x+2\Rightarrow2.5\ge2\cdot0+2 \\ 2.5\ge2 \end{gathered}`
`\begin{gathered} f(x)\leq-4x+3\Rightarrow2.5\leq-4\cdot0+3 \\ 2.5\leq3 \end{gathered}`
`\begin{gathered} f(x)\leq6x+5\Rightarrow2.5\leq6\cdot0+5 \\ 2.5\leq5 \end{gathered}`

Then the point (0,2.5) is a solution for the three equations and is the answer of the problem