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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Equations & Inequalities

7.19. Solution: Using the Method of Signs we can easily solve the
inequality w4 − 15w ≥ 0 in exactly the same way we did in Exercise
7.18.
Thus,
w4 − 15w ≥ 0 has solution w ≤ 0 or w ≥ To continue the analysis, we need to break our argument down into
cases.

Case 1. w ≥ in particular w is positive.
Since w > 0, w4 − 15w < w4. Take the square root of both sides we
get This implies that since we are adding/subtracting a smaller number. But this means, since both numerator and denominator are positive.

Case 2. w ≤ 0. Actually, w ≠0 since w appears in the denominator.
Since w < 0, w4 < w4−15w, since we are subtracting a negative number,
the result will be larger. Taking roots, we get Therefore, This means, since, you’ll recall, we are assuming in this case that w < 0.

Summary.
1. For any w ≥  In this case, there are four solutions for x. Ouch!

2. For any w < 0, only In this case, there are two solutions for x. That was ugly!

7.20. Solutions:
(a) Solve for x: |x + 3| < 8. given from (16) add −3 to all sides

Presentation of Solution: (b) Solve for x: |4x + 9| ≤ 1. given from (16) add −9 to all sides multiply all sides by 1/4

Now, reducing to lowest terms we get the . . .
Presentation of Solution: (c) Solve for x: |2 − 7x| ≤ 3. given from (16) add −2 to all sides multiply all sides by −1/7

or In the last step we have multiplied both sides by a negative
number, this will reverse the direction of the inequality!
Presentation of Solution: 7.21. Solution to (a) Solve for x: |9x − 2| ≥ 3.
|9x − 2| ≥ 3
Use (17) to split the inequality! upper inequality lower inequality add 2 both sides add 2 both sides divide by 9 divide by 9 solution set solution set

Now, join the solutions!
Solution Set Presentation of Solution: Solution to (b) Solve for x: |2 − 3x| > 6.
|2 − 3x| > 6

Use (17) to split the inequality! upper inequality lower inequality add −2 add −2 divide by −3 divide by 5 solution set solution set

Now, join the solutions!
Solution Set Presentation of Solution: Comment: Hopefully, you understand when to include the endpoints
and when not to include them in your solution set, and, most importantly,
how do denote the inclusion/exclusion of the endpoints.

Solution to (c) Solve for x:  Use (17) to split the inequality! upper inequality lower inequality multiply by 6 multiply by 6 add −12 add −12 divide by 9 divide by 9 solution set solution set

Now, join the solutions!
Solution Set Presentation of Solution: 