Example 1: Solve 2(3x + 4)
This equation can be written as: 2 x (3x) + 2 x (4) or 2(3x) + 2(4)
In both cases, we multiply 2 by each single term within brackets.
2(3x + 4) = 6x + 8
If the question asked us to simplify the equation, that would be the answer: 6x + 8
But the question asked us to solve the equation. That means we must put the equation equal to zero and solve for x.
6x + 8 = 0
6x = -8
x = -8 / 6
x = -1.33
Note that we need to solve for x. That means we need to move all non-x values to the opposite side of the equation. When we move +8 over to the other side, it becomes -8. We also need to remove the ‘6’ from the x.
Because it is attached to the x and not alone (like the 8 was), moving the 6 over to the other side means we must divide -8 by that value. With a calculator to hand, we learn that -8 divided by 6 gives us -1.33.
Let’s look at another example.
Example 2: Solve 6 (2a/6) = 4
Another rule of multiplication concerns that 6 behind the brackets.
6 is the same as 6/1. Anything divided by 1 is the same as itself. So, 1 divided by 1 is 1; 7 divided by 1 is 7 etc. The reason this rule is important is because it helps us understand how to answer multiplying an integer (6) by an algebraic fraction (2a/6).
6/1 (2a/6) = 4
Here, then, we can see that the top-half of each fraction may be multiplied out.
6 x 2a = 12a
In turn, we can now divide 12a by 6, to give us 2a. The equation now becomes:
2a = 4
a = 4/2
a = 2
Example 3:
Simplify 3/5+ 1/3
When adding fractions, it’s important to remember to cross-multiply each term.
Let’s flesh this example out to see what we mean.
3/5 + 1/3 = 3(3)/(5(3)) + 1(5)/(3(5)) = 9/15 + 5/15 = 14/15
Because the denominator value (bottom fraction values) is common (both are 15), we only need to now add the numerators (top fraction values).
Example 4:
Simplify 2/(7/4)
Rule: dividing by a fraction is the same as multiplying the numerator (top value, or 2) by the denominator of the dividing fraction (4).
2 / (7/4) = 2(4)/7 = 8/7 = 11/7
Remember the rule for top-heavy fractions. In this case 8/7 is the same as 7/7 + 1/7. Anything divided by itself is 1, so our answer simplifies to 1 and 1/7.
Example 5:
Rule: if the numerator is a fraction, it is the same thing as multiplying both denominators together. Let’s put this rule into action.
(1/4) / 2 = 1/(2 (4)) = 1/8
You can see that we’ve multiplied both fraction denominators. Again, this rule only applies when the numerator of the overall fraction is also a fraction.
Example 6:
Rule – subtracting terms in fractions is the same when the order is reversed.
(3-7) / (2-1) = (7-3) / (1-2)
No matter which version of the fraction we use, the answer is the same: -4.
That completes part I of our ultimate ASVAB algebra study guide. Check back to our ASVAB blog soon for parts II and III, exploring even more of the fundamental rules about algebra that you need to know.