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Solve the equation and inequalities. Write the solution sets in interval notation if possible.
(a)
(b)
©
(d)
(e)

Ask by Pope French. in the United States
Mar 19,2025

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(a) or
(b)
©
(d) or
(e) or

Solución

(a) Solve
For a product to equal zero, one of the factors must be zero.
  1. Set the first factor equal to zero:
  2. Set the second factor equal to zero:
Solution set:
(b) Solve
A product is negative when the factors have opposite signs. The critical points are and . These divide the real line into three intervals.
  1. Interval :
    Choose :
    Product: (positive).
  2. Interval :
    Choose :
    Product: (negative).
  3. Interval :
    Choose :
    Product: (positive).
The inequality holds when the product is negative.
Solution set:
© Solve
This inequality is similar to part (b), but includes the points where the expression equals zero. From part (a), these points are and .
Solution set:
(d) Solve
Here, we need the product to be positive. From our sign analysis in part (b):
  • For , the product is positive.
  • For , the product is positive.
Solution set:
(e) Solve
This inequality requires the product to be positive or zero. In addition to the positive intervals from part (d), we include the zeros at and .
Solution set:

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Bonus Knowledge

To solve the equation and inequalities you’ve provided, let’s first find the roots of the equation .
(a) Set each factor to zero:
  1. →
  2. →
The solutions to the equation are and . In interval notation, this can be expressed as .
(b) Now, let’s solve the inequality . We analyze the intervals determined by the roots:
  • Test intervals: , , and .
  1. Choose a number from , say :
    → doesn’t satisfy.
  2. Choose from , let’s try :
    → satisfies.
  3. Choose from , let :
    → doesn’t satisfy.
Thus, the solution for is: .
© For , we include the endpoints:
The solutions are .
(d) For , we also consider the intervals:
Using previous tests, it satisfies in:
  • The intervals and .
Therefore, the solution here is: .
(e) Finally, for :
The intervals where it is greater or equal to zero, including the roots, gives us:
  • The solution is: .
To summarize:
  • (a)
  • (b)
  • ©
  • (d)
  • (e)

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