need help with abs. inequalitiy (1 Viewer)

[coyazayo]

Q.:solve: |2x-6|<x+3

 

[InteGrand]

Q.:solve: |2x-6|<x+3

We have two cases.

Case 1. 2x – 6 ≥ 0 (i.e. x ≥ 3)

In this Case, we have |2x – 6| = 2x – 6. So the inequation is

2x – 6 < x + 3, x ≥ 3

⇒ x < 9, x ≥ 3.

So 3 ≤ x < 9 is one part of the solution.

Case 2. 2x – 6 < 0 (i.e. x < 3)

In this Case, we have |2x – 6| = –(2x – 6) = 6 – 2x. So the inequation is

⇒ 6 – 2x < x + 3, x < 3

⇒ 3x > 3, x < 3

⇒ x > 1, x < 3.

So 1 < x < 3 is the other part of the solution.

Combining these two parts of the solution, the overall solution is 1 < x < 9.

 

[braintic]

Alternatively:

Solve the equation |2x-6|=x+3 first.

Then graph y=|2x-6| and y=x+3 on the same axes, showing the x values of the points of intersection found in your algebraic solution.

Then use the graph to solve the inequality.

It is essentially the same thing, but many people find it less confusing.