Math CentralQuandaries & Queries


Question from ryan, a student:

Good day!
We we're given this assignment in algebra concerning linear inequalities with absolute value.
In this number I'm confused how to isolate the absolute value, considering there is a variable in the absolute value and there is a variable in the outside.

2+5 |4-3t|+t ≤ -3-4t

Hi Ryan,

|4 - 3t| is either 4 - 3t or it's -(4 - 3t) and I would use this to split the problem into two inequalities right from the start. To be more precise

absolute value.

4 - 3t ≥ 0 is equivalent to t ≤ 4/3 and 4 - 3t < 0 is equivalent to 4/3 < t and hence

absolute value.

So now separate the problem into two situations.

Part 1: t ≤ 4/3

In this case |4 - 3t| = 4 - 3t so the inequality becomes

2 + 5 (4-3t)+ t ≤ -3 -4t

Solve for t. To be a solution of the original problem t must satisfy both the inequality you just obtained and the inequality t ≤ 4/3.

Part 2: t < 4/3

In this case you have |4 - 3t| = -(4 - 3t) so substitute this into the original equation and proceed as in part 1.

Write back if you have any difficulties in completing the solution.


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