Solving Equations & Inequalities
7.19. Solution: Using the Method of Signs we can easily solve the
inequality w^{4} − 15w ≥ 0 in exactly the same way we did in Exercise
7.18. Thus,
w^{4} − 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, w^{4} − 15w < w^{4}. 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, w^{4} < w^{4}−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: