# How do you write the inequality and solve given "twice a number increased by 3 is less than the number decreased by 4"?

Jun 2, 2017

See a solution process below:

#### Explanation:

To write the inequality we will start by defining "a number" as $n$.

Then, "twice a number" can be written as $2 n$.

"increased by 3" becomes: $2 n + 3$

"is less than" turns it into" $2 n + 3 <$

"the number decreased by 4" is: $n - 4$

Putting this together gives the inequality:

$2 n + 3 < n - 4$

To solve, first subtract $\textcolor{red}{3}$ and $\textcolor{b l u e}{n}$ from each side of the inequality to solve for $n$ while keeping the inequality balanced:

$- \textcolor{b l u e}{n} + 2 n + 3 - \textcolor{red}{3} < - \textcolor{b l u e}{n} + n - 4 - \textcolor{red}{3}$

$- \textcolor{b l u e}{1 n} + 2 n + 0 < 0 - 7$

$\left(- \textcolor{b l u e}{1} + 2\right) n < - 7$

$1 n < - 7$

$n < - 7$

Jun 2, 2017

Any number less than $- 7$ will make this inequality true.

$x < - 7$

#### Explanation:

Let the number be $x$

Write the left side as maths first:

$\textcolor{red}{\text{Twice a number ") color(blue)(" increased by 3 ") color(magenta)("is less than}}$

$\textcolor{red}{2 x} \textcolor{w h i t e}{w w w w w w w w w w w w w} \textcolor{b l u e}{+ 3} \textcolor{w h i t e}{w w w w w w w} \textcolor{m a \ge n t a}{<}$

Now do the same for the right side..

color(red)(2x)" " color(blue)(+ 3 ) " "color(magenta)( <)" "color(green)("the number decreased by 4"

color(red)(2x)" " color(blue)(+ 3 ) " "color(magenta)( <)" "color(green)(x-4)

We have the inequality, now solve it:

$2 x + 3 < x - 4$

$2 x - x < - 4 - 3$

$x < - 7$