# How do you solve "quotient of three times a number and 4 is at least -16" and graph the solution on a number line?

Nov 7, 2017

See a solution process below:

#### Explanation:

Let's call "a number": $x$

"The quotient" is the result of division.

In this problem, the numerator is: $3 x$

The denominator is: $4$

So we can write:

$\frac{3 x}{4}$

"is at least" means this is an inequality and specifically a $\ge$ inequality.

So, we can continue to write:

$\frac{3 x}{4} \ge$

And we can finish the inequality as:

$\frac{3 x}{4} \ge - 16$

To solve this, multiply each side of the inequality by $\frac{\textcolor{red}{4}}{\textcolor{b l u e}{3}}$ to solve for $n$ while keeping the inequality balanced:

$\frac{\textcolor{red}{4}}{\textcolor{b l u e}{3}} \times \frac{3 x}{4} \ge \frac{\textcolor{red}{4}}{\textcolor{b l u e}{3}} \times - 16$

$\frac{\cancel{\textcolor{red}{4}}}{\cancel{\textcolor{b l u e}{3}}} \times \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{3}}} x}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}}} \ge - \frac{64}{3}$

$x \ge - \frac{64}{3}$

To graph this we will draw a vertical line at $- \frac{64}{3}$ on the horizontal axis.

The line will be a solid line because the inequality operator contains an "or equal to" clause.

We will shade to the right side of the line because the inequality operator also contains a "greater than" clause:

graph{x>=-64/3 [-30, 30, -15, 15]}

Nov 7, 2017

$x \ge - 21 \frac{1}{3}$
To graph this on a number line, you would make a solid dot on the point $\left(- 21 \frac{1}{3}\right)$, with the line moving to the right ($\rightarrow$)

#### Explanation:

First, let's analyze what each word means.

"quotient ($\div$) of three times a number ($3 x$) and four ($+ 4$) is at least -16 ($\ge - 16$)"

Now take out the numbers.

$3 x \div 4 \ge - 16$

Now to find the possibilities of $x$, balance the inequality.

$3 x \div 4 \ge - 16$ Multiply both sides by 4.
$3 x \ge - 64$ Divide both sides by 3.
$x \ge - 21 \frac{1}{3}$

To graph this on a number line, you would make a solid dot on the point $\left(- 21 \frac{1}{3}\right)$, with the line moving to the right ($\rightarrow$)