# How do you write an inequality and solve given "three fourths of a number decreased by nine is at least forty two"?

Feb 26, 2017

The inequality is: $\frac{3}{4} n - 9 \ge 42$

The solution is: $n \ge 68$

#### Explanation:

First, let's call "a number" - $n$

"three fourths of a number" then can be written as:

$\frac{3}{4} n$

This, "decreased by nine" then can be written as:

$\frac{3}{4} n - 9$

"is at least" is the same as $\ge$ so we can now write:

$\frac{3}{4} n - 9 \ge$

and what it is "at least" is "forty two" so:

$\frac{3}{4} n - 9 \ge 42$

To solve this we first add $\textcolor{red}{9}$ to each side of the inequality to isolate the $n$ term while keeping the inequality balanced:

$\frac{3}{4} n - 9 + \textcolor{red}{9} \ge 42 + \textcolor{red}{9}$

$\frac{3}{4} n - 0 \ge 51$

$\frac{3}{4} n \ge 51$

Now, we 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:

$\frac{\textcolor{red}{4}}{\textcolor{b l u e}{3}} \times \frac{3}{4} n \ge \frac{\textcolor{red}{4}}{\textcolor{b l u e}{3}} \times 51$

$\frac{\cancel{\textcolor{red}{4}}}{\cancel{\textcolor{b l u e}{3}}} \times \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{3}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}}} n \ge \textcolor{red}{4} \times 17$

$n \ge 68$

Feb 26, 2017

Without any punctuation there are different interpretations which lead to different solutions:

$\frac{3}{4} x - 9 \ge 42 \text{ or } \frac{3}{4} \left(x - 9\right) \ge 42$

$x \ge 56 \text{ or } x \ge 65$

#### Explanation:

Let's use some punctuation so show exactly what is meant:
note how the placement of a comma gives a different meaning.

$\textcolor{b l u e}{\text{three fourths of a number"), color(red)(" decreased by 9") color(limegreen)(" is at least 42}}$

Let the number be $x$

$\textcolor{b l u e}{\frac{3}{4} x} \textcolor{red}{- 9} \textcolor{\lim e g r e e n}{\ge 42}$

Solving gives:

$\frac{3}{4} x \ge 51$

$\frac{4}{3} \times \frac{3}{4} x \ge 51 \times \frac{4}{3}$

$x \ge 56$

However, if we interpret it differently we could have:

$\textcolor{b l u e}{\text{three fourths of,") color(red)(" a number decreased by 9,") color(limegreen)(" is at least 42}}$

color(blue)(3/4) color(red)((x-9)) color(limegreen)( >= 42

Solving gives:

$\frac{4}{3} \times \frac{3}{4} \left(x - 9\right) \ge 42 \times \frac{4}{3}$

$x - 9 \ge 56$

$x \ge 65$