What is the solution set of |x-2|(less than or equal to) 4 in interval notation?

Question

Having algebra problems. Please help 

Answer 1

| x - 2 | ≤ 4

 

 

We know that the absolute value of a negative number is always positive.  The same goes for the absolute value of a positive number.

 

We set two separate inequalities.

 

 

x - 2 ≤ 4               and              x - 2 ≥ -4

 

 

Solve the inequalities.

 

x ≤ 6                   and                 x ≥ -2

 

 

The solution set is all real numbers between -2 and 6, including -2 and 6.  In interval notation,

 

[-2, 6]

Answer 2

| x - 2 | ≤ 4

 

Equation/Inequalities containing absolute value must be solved twice: once assuming the argument is positive and once assuming the argument is negative.

 

1. assuming x - 2 ≥ 0

| x - 2 | ≤ 4

x - 2 ≤ 4          add 2 to both sides

x ≤ 2

 

2. assuming x - 2 < 0

| x - 2 | ≤ 4

-(x - 2) ≤ 4        distribute the -1

-x + 2 ≤ 4          subtract 2 from both sides

-x ≤ 2                multiple both sides by -1 (order changes)

x ≥ -2

 

[-2, 2]