1. ## 3 questions

Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the intervals separated by the "union" symbol, U. If needed enter as infinity and as -infinity .

Find the quotient and remainder using long division for

Simplify the expression

2. Originally Posted by wannous
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the intervals separated by the "union" symbol, U. If needed enter as infinity and as -infinity .

Find the quotient and remainder using long division for

Simplify the expression

For Q.3 note that

$7x^3 - 8x^2 - 12x = x(x-2)(7x+3)$

and

$6x^2 - 16x + 8 = 2(x - 2)(3x - 2)$.

So $\frac{7x^3 - 8x^2 - 12x}{6x^2 - 16x + 8} = \frac{x(x-2)(7x+3)}{2(x - 2)(3x - 2)}$

$= \frac{x(7x + 3)}{2(3x - 2)}$.

Therefore $f(x) = x(7x + 3)$ and $g(x) = 2(3x - 2)$.

3. Hello wannous
Originally Posted by wannous
Solve the following inequality. Write the answer in interval notation.
Note: If the answer includes more than one interval write the intervals separated by the "union" symbol, U. If needed enter as infinity and as -infinity .

To solve this type of inequality, find the values of $x$ that make each factor zero, and then look at their signs as $x$ moves along the number line from left to right, through each of these 'zero values' in turn.

Now the 'zeros' occurs when $x =4$ and $x=5$. So we look at what happens when:

• $x<4$

• $4

• $5

First, if $x<4,\, (x-5)$ is negative and $(x-4)$ is also negative; and $-^{ve}\times -^{ve} = +^{ve}$. So $(x-5)(x-4)>0$ in this range.

Next, if $4 is $-^{ve}$ and $(x-4)$ is $+^{ve}$; $-^{ve}\times+^{ve} = -^{ve}$. So $(x-5)(x-4) < 0$ in this range.

Finally, if $x>5$, we get $+^{ve}\times+^{ve}=+^{ve}$. So $(x-5)(x-4)>0$ in this range.

So the values we want are $4\le x \le 5$, or in interval notation $[4,5]$.

Find the quotient and remainder using long division for

See the attached image.

So the quotient is
$x^2-4x+1$ and the remainder is $-5$.