Solve the inequality
|x-3|>1

2 < x < 4

Short explanation

(in case you just needed a little help):

|x - 3| > 1 implies 4 things

(assuming x-3 is positive):

1. x - 3 > 1, so x > 4

2. 3 - x < 1, so x > 2

(One of these is true)

(Assuming x-3 is negative):

3. x-3 < 1, so x< 4

4. 3-x > 1, so x <2

(One of these is true)

The only inequalities from each pair that make sense here are

x>2 and x <4.

2 < x < 4.

LONGER EXPLANATION

(in case your teacher didn’t teach the subject well):

|x - 3| > 1 implies

(assuming x-3 is positive):

1. x - 3 > 1

2. 3 - x < 1

Why? Because the absolute value takes any negative or positive value and spits out the positive version of it. If x-3 is positive and |x - 3| > 1,

then x-3 > 1.

Also if x-3 is positive, 3-x must be negative so 3-x < 1 since all negative numbers are less than 0 which is less than 1.

Now we just work out values of x from the inequalities we just discovered.

1. x-3 > 1:

x > 4 (Add 3 to both sides)

2. 3-x < 1:

-x < -2 (subtract 3 from both sides)

x > 2 (multiplying both sides by (-1) will also flip the inequality).

That was all if x - 3 was positive. But if x-3 is negative then:

3. x-3 < 1, so x < 4

4. 3-x > 1. so x < 2

So we have 2 pairs of inequalities but only 1 inequality in each pair can be true:

1. x>2

2. x>4

3. x<4

4. x<2

The only inequalities from each pair that make sense here are #1 and #3

x>2 and x <4.

#2 doesn’t make sense because if x>4 (first pair) then that contradicts both inequalities in the second pair x<4 and x<2, and we know one of them has to be true.

#2 doesn’t work, so #1 must be true: x>2. Now we need to see which inequality is true in the second pair: x<4 or x < 2.

If x>2 then that contradicts what we found with #1. x can’t be less than 2 at the same time. So #4 can’t be true and it must be the case that #3 is true so x<4. So we found a pair that works and have a solution!

x<4 and x>2

So 2 < x < 4.

43

step-by-step explanation:

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