Question

0> -2x^2+4x+4Solve each inequality by graphing. Sketch it.

Answer 1

To solve the inequality we need to find the x-values that are the roots of the quadratic equation, let's use the quadratic formula:

`\begin{gathered} \text{For an equation in the form:} \\ ax^2+bx+c=0 \\ The\text{ quadratic formula is:} \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \text{Then a=-2, b=4 and c=4} \\ x=\frac{-4\pm\sqrt[]{4^2-4(-2)(4)}}{2(-2)} \\ x=\frac{-4\pm\sqrt[]{16+32}}{-4} \\ x=\frac{-4\pm\sqrt[]{48}}{-4} \\ x=(-4\pm6.93)/(-4) \\ \text{Then} \\ x1=(-4+6.93)/(-4)=(2.93)/(-4)=-0.732 \\ x2=(-4-6.93)/(-4)=(-10.93)/(-4)=2.732 \end{gathered}`

Now, let's try values less or greater than these roots:

If x=-1:

`\begin{gathered} 0>-2(-1)^2+4(-1)+4 \\ 0>-2\cdot1-4+4 \\ 0>-2\text{ This is right, then number less than -0.732 are solutions of the inequality} \end{gathered}`

Now let's try x=3:

`\begin{gathered} 0>-2(3)^2+4(3)+4 \\ 0>-2\cdot9+12+4 \\ 0>-18+16 \\ 0>-2\text{ This is correct two, then the values greater that 2.732 are solutions to the inequality too} \end{gathered}`

Then, the graph of the inequality is:

The red-shaded area are the solution to the inequality, then in interval notation we have:

`(-\infty,-0.732)\cup(2.732,\infty)`

In builder notation it would be:

`x|x<-0.732orx>2.732`