Exercises Involving Absolute Value Equations

1.  The absolute value equation |x - 2| = 5 can be looked at geometrically as the value of x such that the Distance from x to 2 is 5.

For exercises 3 through 6, determine whether the statement is true or false and explain your reasoning.

3.  The equation |ax - b| = c where a, b, and c are constants has either no solution, one solution, or two solutions.

True, if c < 0, then there are no solutions, if c = 0 there is one solution, and if c > 0 there are two solutions.

5.  The expression |4x - 2| is equal to the expression |4x + 2| for all values of x.

False, the absolute value does not change subtraction into addition.  For example if x = 1 then |4(1) - 2|  =  2 and |4(1) + 2|  =  6.

For exercises 7 through 33, solve for x.  If there is no solution, state that there is no solution.

7.  |x| = 6

x = -6  or x = 6

9.  |x| = -2

No solution

11.   |x + 5|  =  1

x = -6  or  x = -4

13.  |x - 1| = 0

x = 1

15.  |3x - 5|  =  2

x = 1  or  x = 7/3

17.  |5 - 2x|  =  3

x = 1  or  x = 4

19.  |2x - 4|  + 3 =  5

x = 1  or  x = 3

21.  |2x + 1| + 2  =  2

x = -1/2

23.  |0.5x + 0.2|  =  0.4

x = -6/5  or  x = 2/5

25. 

x = -3/8   or   x = 9/8

27.  |2(x - 3) + 4|  =  5

x = -3/2  or  x = 7/2

29.  3|x - 4| + 2  =  6

x = 8/3  or  x = 16/3

31.  |x|  =  |2x - 4|

x = 4/3  or  x = 4

33.  |3x + 1|  =  |2x - 3|

x = -4  or  x = 2/5

35.  The elevations of Greenpark and Forksmith differ by 1200 feet.  Use an absolute value equation and the fact that Forksmith's elevation is 2304 ft to find the possibilities for the elevation of Greenpark.

|x - 2304|  = 1200

so

x = 1104  or  x = 3504

Greenpark's elevation is either 1104 ft or 3504 ft.

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