SOLUTION: Solve the equation |x+1| + |2x-3| = 8.

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Question 1184798: Solve the equation |x+1| + |2x-3| = 8.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


For x<-1, abs%28x%2B1%29=-x-1; for x>-1, abs%28x%2B1%29=x%2B1.

For x<3/2, abs%282x-3%29=-2x%2B3; for x>3/2< abs%282x-3%29=2x-3.

To solve the equation then, we need to consider three intervals of possible values for x: (1) x less than -1; (2) x between -1 and 3/2; and (3) x greater than 3/2.

Solve the equation on each of those intervals, remembering that any "solution" we get has to be in the interval we are working in.

(1) x less than -1

abs%28x%2B1%29%2Babs%282x-3%29=-x-1-2x%2B3

-x-1-2x%2B3=8
-3x%2B2=8
-3x=6
x=-2

Since our solution x=-2 is in the interval x<-1, it is a solution; we can easily check that:

abs%28x%2B1%29%2Babs%282x-3%29=abs%28-1%29%2Babs%28-7%29=1%2B7=8

(2) x between -1 and 3/2

abs%28x%2B1%29%2Babs%282x-3%29=x%2B1-2x%2B3

x%2B1-2x%2B3=8
-x=4
x=-4

That solution is NOT in the interval we are working in; it is not a solution.

(3) x greater than 3/2

abs%28x%2B1%29%2Babs%282x-3%29=x%2B1%2B2x-3

x%2B1%2B2x-3=8
3x=10
x=10%2F3

That solution is in the interval we are working in, so it is a solution.

abs%28x%2B1%29%2Babs%282x-3%29=abs%2813%2F3%29%2Babs%2811%2F3%29=13%2F3%2B11%2F3=24%2F3=8

ANSWERS: x=-2; x=10/3

A graph, showing abs(x+1)+abs(2x-3) (red) = 8 (green) at x=-2 and at x=10/3....

graph%28400%2C400%2C-5%2C5%2C-2%2C10%2Cabs%28x%2B1%29%2Babs%282x-3%29%2C8%29