a) Given the equation
[tex]5x^2-x^3=-x^2+3x+4[/tex]First, simplify the equation
[tex]\begin{gathered} 5x^2-x^3+x^2=-x^2+3x+4+x^2 \\ 6x^2-x^3=3x+4 \\ 6x^2-x^3-3x=3x+4-3x \\ 6x^2-x^3-3x=4 \\ 6x^2-x^3-3x-4=4-4 \\ 6x^2-x^3-3x-4=0 \\ \operatorname{Re}-\text{order} \\ -x^3+6x^2-3x-4=0 \end{gathered}[/tex]Then we graph the equation
The solutions are the values of x for which y = 0. These are:
[tex]\begin{gathered} x=-0.58 \\ x=1.29 \\ x=5.29 \end{gathered}[/tex]Answer a: x = -0.58, 1.29, 5.29
b) Given the inequality
[tex]5x^2-x^3\leq-x^2+3x+4[/tex]Solve for x using the graph
So, the solution is:
[tex]\begin{gathered} -0.58\leq x\leq1.29 \\ or \\ x\ge5.29 \end{gathered}[/tex]In interval notation:
[tex]\lbrack-0.58,1.29\rbrack\cup\lbrack5.29,\infty)[/tex]Answer b:
[tex]\lbrack-0.58,1.29\rbrack\cup\lbrack5.29,\infty)[/tex]
