# Solve the inequality x2 + 9x – 10 < 0 ?

May 1, 2018

The interval $\left(- 10 , 1\right)$. This means all numbers between -10 and 1, excluding both the limits.

#### Explanation:

${x}^{2} + 9 x - 10 < 0$
The procedure to solve a polynomial inequality is to first factorize it.
$\implies {x}^{2} + 10 x - x - 10 < 0$
$\implies x \left(x + 10\right) - 1 \left(x + 10\right) < 0$
$\implies \left(x - 1\right) \left(x + 10\right) < 0$

The second step is to find the zeroes of the polynomial after factorization. You will understand why when we get to the next step.
Clearly, when $x = 1 \mathmr{and} x = - 10$, the left hand side is equal to zero.
We now plot the points (1) and (-10) on a number line. This divides the line into 3 distinct parts: the part less than -10(call this part one, or P1), one part between -10 and 1(P2), and the last being the part greater than 1(P3).

Let us now put a value of x greater than $x = 1$. Suppose we plug in two.$\left(2 - 1\right) \left(2 + 10\right) = 12$ Observe that the sign of the value we get from the polynomial when $x = 2$ is positive.

2 is in P3. So we mark P3 as POSITIVE . This means all numbers in P3(all numbers greater than 1) result in a postive value of the polynomial. Let us now set the signs for P2 and P1. P2 will be negative and P1 will be positive. This is a rule of the method: once we figure out the sign of a part, we alternate the signs for the remaining parts.

We now know that all values in P3 and P1 result in positive numbers. We also know that P2 will give negative values.
Clearly, only negative values will satisfy the condition that the polynomial is less than 0. Thus the answer is the values of x that result in negative values of the polynomial: P2.

Recollect that P2 refers to the numbers between -10 and 1. So the solution is all numbers between -10 and 1, excluding both. This is because -10 and 1 result in 0, while the question asks values below 0. Mathematically, this interval is called $\left(- 10 , 1\right)$.

I know this may seem confusing; that's because it is! Ask your teacher to explain the Wavy Curve Method(that's what this is called, by the way).

May 1, 2018

$- 10 < x < 1$

#### Explanation:

$\text{factor the quadratic}$

$\Rightarrow \left(x + 10\right) \left(x - 1\right) < 0$

$\text{find the zeros by solving } \left(x + 10\right) \left(x - 1\right) = 0$

$\Rightarrow x = - 10 \text{ or } x = 1$

$\text{since "a>0" then } \bigcup$

$\Rightarrow - 10 < x < 1$

$x \in \left(- 10 , 1\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$
graph{x^2+9x-10 [-20, 20, -10, 10]}