Concept Byte: Rational Inequalities

For Use With Lesson 8-6

TECHNOLOGY

Consider the rational inequality x 4 − x < 3 . fraction x , over 4 minus x end fraction . less than 3 .

Activity 1

1. Enter y 1 = x 4 − x y sub 1 , equals . fraction x , over 4 minus x end fraction  and y 2 = 3 y sub 2 , equals 3  in your graphing calculator. Graph the functions using the settings below. Use the calculator's INTERSECT feature to find where the two functions are equal. Use the graph to find the solution of the inequality x 4 − x < 3 . fraction x , over 4 minus x end fraction . less than 3 .

   
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Activity 2

Using the functions you entered in Activity 1, set up the table as shown.

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2. Scroll to x-values less than 3. Do they make the inequality true?
3. Scroll to x-values greater than 4. Do they make the inequality true?
4. What happens to the inequality when x = 3 ? x equals 3 question mark  When x = 4 ? x equals 4 question mark
5. Change Δ Tbl cap delta cap tbl  to 0.1. Investigate the inequality between x = 3 x equals 3 and x = 4. x equals 4.
6. Make a conjecture about the solution of the inequality based on your results in Step 2–5.

Activity 3

Now use algebra to solve the inequality. You can multiply both sides of a rational inequality by the same algebraic expression just as you have done with equations. But you must keep in mind the properties of inequalities. Consider the first step, multiplying each side by ( 4 − x ) . open 4 minus x close .

x 4 − x < 3 ( 4 − x ) x 4 − x < ( 4 − x ) 3 table with 2 rows and 1 column , row1 column 1 , fraction x , over 4 minus x end fraction . less than 3 , row2 column 1 , open , 4 minus x , close . fraction x , over 4 minus x end fraction . less than . open , 4 minus x , close . 3 , end table

Depending on whether the factor ( 4 − x ) open 4 minus x close  is positive or negative, there are two possible solutions to the inequality.

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