Chapter 6: Rationals: Multiplication, Division, and Applications


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!)



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]






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.