Question

graph the solution to the following inequality on the number line.
(-x^{2}+2x < - 3)

Explanation:

Step1: Rearrange the inequality

First, rewrite $-x^{2}+2x < - 3$ as $-x^{2}+2x + 3<0$. Multiply both sides by - 1 to get $x^{2}-2x - 3>0$ (note that the inequality sign flips when multiplying by a negative number).

Step2: Factor the quadratic

Factor $x^{2}-2x - 3$ as $(x - 3)(x+1)>0$.

Step3: Find the roots

Set $(x - 3)(x + 1)=0$. The roots are $x=3$ and $x=-1$.

Step4: Determine the solution intervals

We consider the intervals $(-\infty,-1)$, $(-1,3)$ and $(3,\infty)$. Test a value from each interval in the inequality $(x - 3)(x + 1)>0$. For $x=-2$ (in $(-\infty,-1)$), $(-2 - 3)(-2+1)=(-5)\times(-1) = 5>0$. For $x = 0$ (in $(-1,3)$), $(0 - 3)(0 + 1)=(-3)\times1=-3<0$. For $x=4$ (in $(3,\infty)$), $(4 - 3)(4 + 1)=1\times5 = 5>0$. So the solution of the inequality is $x<-1$ or $x>3$.

Answer:

On the number - line, we use an open circle at $x=-1$ and draw an arrow to the left, and use an open circle at $x = 3$ and draw an arrow to the right.