how to solve compound inequality with multiple variables

Question

5p-10<5-3c<7c

 

Answer 1

Kimberly,

 

The inequalities contain 2 variables, p & c.  So think of a 2-domensional space and graphing the inequalities on that space to see what portions of that space satisfy them both.

 

Your compound inequality has 3 pieces to it separated by the inequality operators.  And the rightmost 2 pieces only involve c so you can solve for c.

 

(5 - 3c) < 7c  ,  so 5 < 10c , and (1/2) < c, so on your graph c can be anywhere in open interval (1/2, ∞).  Note that c = 1/2 itself does not satisfy the inequality and must be left out.

 

Now solve the 2 leftmost parts to get p

(5p - 10) < (5 - 3c) or 5p < (15 - 3c), and  p < (3 -3c/5) so plot this line.

Since the c-term is negative, the largest p can be is when c is smallest, just a hair to the right of 1/2. Plug in the 1/2 and apply the hair later to p,  so p < 27/10 at c = 1/2.  Also at c = 5, p = 0 so it is a line that goes thru the two points: (1/2, 27/10) and (5, 0).

 

Then all all (c, p) points below that line such that c> 1/2 satisfy the compound inequality.

 

Thus, to solve a compound inequality you separate it into pieces that are more easily solvable and then reassemble those partial solutions into the overall solution.  Graphically plotting them helps to visualize the individual pieces and the whole solution, and avoid getting confused by the details.