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EXAMPLES OF INEQUALITIES WITH ABSOLUTE VALUES

Example 1:

Given   4 |x + 1| ≥ x -1  Solve for x

SOLUTION:

Example 2:

Solve |x + 3| + |x - 1| > |2x + 1| - |x - 2| + 4

Solution:

Find the break-points for each absolute values (expressions between the bars):

For x + 3 = 0, x = -3

For x - 1 = 0, x = 1

For 2x + 1 = 0, x = -1/2

For x - 2 = 0, x = 2

Set up the intervals for -3, -1/2, 1, 2

Interval 1:    x < -3

Interval 2:    -3 < x  < -1/2

Interval 3:    -1/2 < x < 1

Interval 4:    1 < x < 2

Interval 5:    2< x


Evaluate the expressions for each intervals:


For Interval 1
:    x < -3

- (x + 3) - (x - 1) > - (2x +1) - [- (x - 2)] + 4

- x - 3 - x + 1 > - 2x -1 + x - 2 + 4

-2x -2 > -x +1

x < -3

Conclusion for interval 1:  x < -3 (It is true because it is inside the interval)


For Interval 2
:     -3 < x < -1/2

(x + 3) - (x - 1) > - (2x +1) - [- (x - 2)] + 4

x + 3 - x + 1 > - 2x -1 + x - 2 + 4

4 > -x +1

 -3 < x

Conclusion for interval 2:    -3 < x < -1/2  (It is true because it is inside the interval)


For Interval 3:     -1/2 < x < 1

(x + 3) - (x - 1) > (2x +1) - [- (x - 2)] + 4

x + 3 - x + 1 > 2x + 1 + x - 2 + 4

4 > 3x + 3

1 > 3x

 x < 1/3

Conclusion the interval 3:   -1/2 < x < 1/3  (It is true because it is inside the interval)


For Interval 4
:     1 < x < 2

(x + 3) + (x - 1) > (2x +1) - [- (x - 2)] + 4

x + 3 + x - 1 > 2x + 1 + x - 2 + 4

2x + 2 > 3x + 3

x < -1

Conclusion for interval 4:     x < -1 NOT TRUE (It is not true because it is not inside the interval)


For Interval 5
:     2 < x

(x + 3) + (x - 1) > (2x +1) - (x - 2) + 4

x + 3 + x - 1 > 2x + 1 - x + 2 + 4

2x + 2 > x + 7

5 < x

Conclusion for interval 5:   5 < x  (It is true because it is inside the interval)


We also test the inequality for the break-points: x = -3, -1/2, 1 and 2

Inequality does not work for the x = -3, 1 and 2 (not true)

Inequality works only for the x = -1/2 (true)

 

COMBINE:

Conclusion for interval 1:  x < -3
Conclusion for interval 2:    -3 < x < -1/2
Conclusion the interval 3:   -1/2 < x < 1/3
Conclusion for interval 5:   5 < x
x = -1/2

When we combine above inequalities in a number line we get:

FINAL RESULT:  x < 1/3   or   5 < x    (x does not equal to -3)