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

This statement can be expressed algebraically as:

#Rightarrow 10 times x + 5 > 12 times x - 1#

#Rightarrow 10 x + 5 > 12 x - 1#

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

#Rightarrow 10 x - 10 x + 5 > 12 x - 10 x - 1#

#Rightarrow 5 > 2 x - 1#

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

#Rightarrow 5 + 1 > 2 x - 1 + 1#

#Rightarrow 6 > 2 x#

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

#Rightarrow frac(6)(2) > frac(2 x)(2)#

#Rightarrow 3 > x#

#therefore x < 3#

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." #=># #10n+5gt12n-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:

#10n+5gt12n-1=>-2ngt-6#

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

#nlt3#