# Ten times a number increased by 5 is greater than twelve times a number decreased by one. What is the number?

Jun 12, 2017

The number can be any number less than $3$.

#### Explanation:

This statement can be expressed algebraically as:

$R i g h t a r r o w 10 \times x + 5 > 12 \times x - 1$

$R i g h t a r r o w 10 x + 5 > 12 x - 1$

Let's subtract $10 x$ from both sides of the equation:

$R i g h t a r r o w 10 x - 10 x + 5 > 12 x - 10 x - 1$

$R i g h t a r r o w 5 > 2 x - 1$

Then, let's add $1$ to both sides:

$R i g h t a r r o w 5 + 1 > 2 x - 1 + 1$

$R i g h t a r r o w 6 > 2 x$

Now, let's divide both sides by $2$:

$R i g h t a r r o w \frac{6}{2} > \frac{2 x}{2}$

$R i g h t a r r o w 3 > x$

$\therefore x < 3$

Jun 12, 2017

The number is not a fixed numerical value. Instead the number is any number that is less than $3$.

#### Explanation:

The most common math trick is to use a variable to represent an unknown value. Here we have "the number" as our unknown value. Therefore, we

let $n$ = the number in the problem

After we have setup our variable and defined what it represents, we can go ahead and use the variable for its intended purpose. We will convert the words in the problem into the language of mathematics:

"Ten times a number increased by $5$ is greater than twelve times a number decreased by one." $\implies$ $10 n + 5 > 12 n - 1$

Now that we have our inequality, let's move all the variable terms to the left side and all the numerical terms to the right:

$10 n + 5 > 12 n - 1 \implies - 2 n > - 6$

Now, we can divide both sides by $- 2$, switch the inequality sign around, and obtain $n$:

$n < 3$