Chapter 6: Rationals: Multiplication,
Division, and Applications

by
Millard Lewis and John Moore

## Day 7: Solving Equalities and Inequalities Using Rational Numbers

This lesson will be devoted to the solution of equalities and inequalities
over the rational numbers. (This includes integers, right?!?) This solution
process is identical to the one used for equalities and inequalities comparing
integers that you have seen already. (Oh! You haven't seen them? Click here!)

**Equalities**
Say you have the equation: x + 7/8 = 3/8

How would you go about solving this for x?

Since we are working with an equation (one side of the "equals"
sign is equal to the other side), we have to perform the same operation
to both sides to maintain the equality. We are solving for x, so we need
to isolate x. By subtracting 7/8 from both sides, we obtain: x + 0 = 3/8
- 7/8 = -(4/8) = -(1/2) [we still have to reduce our answer!] Therefore,
x = -(1/2).

Here are more examples:

1. (4/3) * r = 6/7 [To solve this equation, we need r by itself. Since
r is multiplied by 4/3, we have to multiply (4/3)*r by the **multiplicative
inverse** of 4/3. (Recall the **multiplicative inverse** from Day 3!)
So, multiplying both sides of the equation by 3/4 yields: 1*r = 9/14. Therefore,
r = 9/14.]

2. y - 5/16 = 1/4 y = 9/16 [add 5/16 to both sides]

3. t/(-5.8) = 7.23 t = -41.934 [multiply both sides by -5.8]

ASK FOR ASSITANCE IF THESE EXAMPLES ARE NOT ENOUGH!!!

__
__**Inequalities**

Inequalities are solved in the same general way as equalities, but the
two sides of the inequality are most likely not equal! In fact, they are
solved the same with the exception of when you multiply or divide both sides
by a negative number. When this happens, change the direction of the inequality
symbol!!!

Example: -(3/7)x > 9/49 [Here, as with equalities, we multiply both
sides of the inequality by -(7/3). Hence, we obtain, x < -(3/7). NOTICE:
The "greater than" sign changes to a "less than" sign!]

More examples:

1. 12.43 + h > 7.8 h > -4.63 [Subtract 12.43 from both sides.]

2. x / -(8/11) < (33/12) x > -2 [This is a tricky one! You have
to multiply both sides by -(8/11). Then the sign also has to be changed.
Be careful!!]

3. t - 1/2 > 0 t > 1/2 [Add 1/2 to both sides.]

A way to represent inequalities that allows you to see what's going on
is to plot them on the number line. Try this GSP
sketch to try it for yourself! Click here
for your worksheet.