### Polynomials & Rational Functions

Many ways problem

## Solution

Fill in the blank spaces so that each row contains:

• an inequality
• a graph that can help solve the inequality
• a solution set of the inequality.
Solution set Graph Inequality
$\left\{x: 2<x<5\right\}$

One way to start might be to think about the solution set on a pair of axes, and then sketching a graph that would fit the constraints.

The graph sketched above is $y = a(x-2)(x-5)$ where $a>0$. For the inequality we could suggest something specific such as $(x-2)(x-5) < 0$, or we could be more general and write $a(x-2)(x-5) < 0$.

What would happen if $a < 0$? How would the graph change? How would the inequality change?

Solution set Graph Inequality
$\left\{x: 0<x≤1\right\}$

Why can’t this solution set satisfy a quadratic inequality?

We want $1$ to be included, but $0$ is not part of the solution. What graphs can we sketch that will satisfy this constraint? Two suggestions are shown here.

Both graphs have asymptotes at the $y$-axis, which gives the correct solution set since the two functions don’t exist when $x=0$.

$y = \dfrac{1}{x^2}-1$ also has an asymptote on the $y$-axis and a root at $x = 1$. Why isn’t the inequality $\dfrac{1}{x^2}-1≥0$ a possible answer?

Solution set Graph Inequality
$\dfrac{x^2+5x-6}{x^2+x-6}≥0$

We want to know where our graph is greater than or equal to zero. To find these sections we need to know the $x$-intercepts and the asymptotes.

The two asymptotes can be found by finding for what values the denominator, $x^2 + x - 6$, equals zero, which is $x = -3$ and $x = 2$.

The graph will cross the $x$-axis when $y = 0$, which means when $x^2 + 5x - 6 =0$. The intercepts are at $x = -6$ and $x = 1$.

Therefore the solution set of $\dfrac{x^2+5x-6}{x^2+x-6}≥0$ is $\left\{x: x≤-6\right\} \cup \left\{x:-3<x≤1\right\}\cup \left\{x: x>2\right\}$.

Solution set Graph Inequality
$x^2 + 6x > 2x+1$

There are several different graphs we could sketch to help us solve $x^2 + 6x > 2x+1$. Two examples are shown below.

Both examples can be sketched reasonably accurately without having to do any calculations. They give us a way to approximate solutions and enable us to see what the solution set will look like, i.e. $\left\{x: a<x<b\right\}$ or $\left\{x: x<a\right\}\cup \left\{x: x>b\right\}$.

What other graphs could be sketched? Are they any more or less helpful than those above?

Finding the intersection points by solving $x^2 + 6x = 2x+1$ gives us a solution set of $\left\{x: x<-2-\sqrt{5}\right\} \cup \left\{x: x> -2 + \sqrt{5} \right\}$.

Solution set Graph Inequality
$\left\{x: x<-5\right\} \cup \left\{x:-2<x<2\right\}$

The solution set implies that we have three values where the graph switches from positive to negative. If we interpret these as roots then we could sketch a cubic.

This is the graph of $y = a(x+5)(x+2)(x-2)$ where $a > 0$, so the inequality will be $a(x+5)(x+2)(x-2) ≤ 0$.

What if the three values are not all roots? If one of them is an asymptote can you still find a graph that fits the solution set?