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in Linear Equations by (15.7k points)

Solve each of the following in equations and represent the solution set on the number line.

|x + a| + |x| > 3, x ϵ R.

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1 Answer

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by (15.2k points)

Given: 

|x + a| + |x| > 3, x ϵ R. 

|x + a| = -(x + a) or (x + a) 

|x| = -x or x 

When |x + a| = -(x + a) and |x| = -x 

Then, 

|x + a| + |x| > 3 → -(x + a) + (-x) > 3

-x -a – x > 3 

-2x – a > 3 

Adding a on both the sides in above equation 

-2x -a + a> 3 + a 

-2x > 3 + a 

Dividing both the sides by 2 in above equation

Multiplying both the sides by -1 in the above equation

Now when, |x + a| = -(x + a) and |x| = x 

Then, 

|x + a| + |x| > 3 → -(x + a) + x > 3 

-x -a + x > 3 

– a > 3 

In this case no solution for x. 

Now when, |x + a| = (x + a) and |x| = -x 

Then, 

|x + a| + |x| > 3 → (x + a) + (-x) > 3 

x + a - x > 3

a > 3 

In this case no solution for x. 

Now when, 

|x + a| = (x + a) and |x| = x 

Then, 

|x + a| + |x| > 3 → (x + a) + (x) > 3 

x + a + x > 3 

2x + a > 3 

Subtracting a from both the sides in above equation 

2x + a – a > 3 – a 

2x > 3 – a 

Dividing both the sides by 2 in above equation

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