four times the quantity of the sum of seven and a number doubled is less than twelve

Answer 1

Best way to solve this problem is to analyze what the words mean, then input that analysis into the form an equation, and finally to solve, graph, and check the equation.

Let's start with the words:

four times

--We're going to multiply something by four

the quantity

--I'd use parentheses for this because we're going to have to distribute the four to multiple terms

of the sum of seven and a number doubled

--So we're adding a seven to 2 times a number (we'll use X since we don't know what the number is)

is less than twelve

--This is our inequality and the right side of our equation

So here's what we've come up with:

4(7+2x) < 12

First we'll distribute the four.

Four times seven is 28; four times 2x is 8x. The inequality and the 12 don't change.

28 + 8x < 12

We want the 28 and the 12 on the same side of the equation so we can calculate them. The 28 is positive, so to eliminate it from the left side of the equation, we subtract (the opposite of addition) 28 from both sides of the equation. 28-28 is zero, so it eliminates from the left; 12 minus 28 is -16. Here's how the equation looks now:

8x < -16

So all we do is divide by the coefficient of x so we can isolate it. The coefficient of x is 8, so dividing 8 on both sides leaves us with our answer:

x < -2

To graph: On a number line, make a tic mark at -2, then an open circle around it. Make a ray starting from that circle and going left signifying an infinite number of solutions that continue to negative infinity.

To check: Substitute a -2 for each instance of x in the original equation, and pretend the inequality sign is an equal sign.

4(7+2[-2])=12

First we'll multiply the negative and positive twos (since they're within a pair of parentheses), which give us a negative four.

4(7-4) = 12

The subtraction (also within a pair of parentheses) comes next. 7-4 is 3.

4(3) = 12

Finally we multiply the 4 and 3 and get 12.

And 12 does indeed equal 12.