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What is the answer? I need help finding it.

What is the answer? I need help finding it.-example-1
User Izzekil
by
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1 Answer

3 votes

Answer:

c = {-2/3, 1}

Explanation:

You want the solutions to the absolute value equation ...

|2c +8| = |10c|

Breakpoints

The equation's domain can be divided into parts that have boundaries where the argument of the absolute value function is zero. Here, those boundaries are ...

2c +8 = 0 ⇒ c = -8/2 = -4

10c = 0 ⇒ c = 0

Piecewise function

Based on these boundary values, we can rewrite the equation into three different equations in three different domains. The absolute value function negates a negative argument, so we can write ...

  • -(2c +8) = -(10c) . . . . for c < -4
  • 2c +8 = -(10c) . . . . for -4 ≤ c < 0
  • 2c +8 = 10c . . . . for 0 ≤ c

c < -4

-(2c +8) = -(10c) . . . . equation definition in this domain

2c +8 = 10c . . . . . . . multiply by -1

8 = 8c . . . . . . . . . . . subtract 2c

1 = c . . . . . . . . . . . . divide by 8; no solution in the given domain

-4 ≤ c < 0

2c +8 = -10c . . . . equation definition in this domain

12c +8 = 0 . . . . . add 10c

c +2/3 = 0 . . . . . divide by 12

c = -2/3 . . . . . . . subtract 2/3

0 ≤ c

2c +8 = 10c . . . . equation definition in this domain

8 = 8c . . . . . . . . subtract 2c

1 = c . . . . . . . . . divide by 8

The solutions to the equation are c = -2/3 and c = 1.

__

Additional comment

These equations are nicely solved by a graphing calculator. The equation can be rewritten to the form f(c) = 0 by subtracting one side from both sides. Then the solutions are the x=intercepts of the graph.

You will notice the breakpoints in the graph at x = -4 and x = 0, where the nature of the relation changes.

What is the answer? I need help finding it.-example-1
User Kxc
by
8.5k points

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