PLZ HELP What is the best interpretation of P(3) =12
2 answers:
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Answer:
b = 12 and 28
Step-by-step explanation:
The absolute value equation |1/2b-8| = |1/4b-1| resolves to a piecewise linear function with three pieces. There are two solutions.
<h3>Domains</h3>The absolute value function on the left has a turning point where its value is zero:
1/2b -8 = 0
b -16 = 0
b = 16
The absolute value function on the right has a turning point where its value is zero:
1/4b -1 = 0
b -4 = 0
b = 4
For b > 16, both absolute value functions are identity functions. In this domain, the equation is ...
1/2b -8 = 1/4b -1
For 4 < b < 16, the function on the left negates its argument, so the equation in this domain is ...
-(1/2b -8) = 1/4b -1
For b < 4, both functions negate their arguments, so the equation in this domain is ...
-(1/2b -8) = -(1/4b -1)
For the purpose of finding the value of b, this is effectively identical to the equation for b > 16. (The value of b does not change if we multiply both sides of the equation by -1.)
<h3>Solutions</h3><u>Domain b < 4 ∪ 16 < b</u>
1/2b -8 = 1/4b -1
2b -32 = b -4 . . . . . . . . multiply by 4
b = 28 . . . . . . . . . . . . add 32-b to both sides
This solution is in the domain of the equation, so is one of the solutions to the original equation.
<u>Domain 4 < b < 16</u>
-(1/2b -8) = 1/4b -1 . . . . equation in this domain
-2b +32 = b -4 . . . . . . multiply by 4
36 = 3b . . . . . . . . . . . add 2b+4 to both sides
12 = b . . . . . . . . . . . . divide by 3
This solution is in the domain of the equation, so is the other solution to the original equation.
<h3>Graph</h3>For the purposes of the graph, we have define the function g(b) to be the difference of the two absolute value functions. The solutions are found where g(x) = 0, the x-intercepts. The graph shows those to be ...
b = 12 and b = 28
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<em>Additional comment</em>
Defining g(b) = |1/2b-8| -|1/4b-1|, we can rewrite it as ...
Then the solutions are the values of b that make g(b) = 0.
Answer:
Step-by-step explanation:
Use cross products to simplify the equation. Multiply the denominator of the second fraction by the numerator of the first fraction, and the denominator of the first fraction by the numerator of the second,
Simplify,
Inverse operations,