Question

Solve the inequality

|3x+4|>6

Answer 1

x > 2/3 or x < -10/3

Explanation:

|3x + 4| > 6

To solve an absolute value inequality of the form |X| > b, where X is an expression, and b is a number, change the inequality into this compound inequality:

X > b or X < -b

Here, X is 3x + 4.

b is 6.

We get the compound inequality:

3x + 4 > 6 or 3x + 4 < -6

3x > 2 or 3x + 4 < -6

x > 2/3 or 3x < -10

x > 2/3 or x < -10/3

Answer 2

`\left(- \infty, -(10)/(3)\right) \cup \left((2)/(3), \infty\right)`

Explanation:

To solve an inequality containing an absolute value:

1. Isolate the absolute value on one side of the equation.
2. Apply the relevant absolute rule.
3. Solve both cases.

Given inequality:

`|3x+4| > 6`

Apply the absolute rule:

`\textsf{If }\:|u| > a,\:a > 0\:\textsf{ then }\:u < -a \:\textsf{ or }\: u > a`

Therefore:

`\begin{aligned}\text{\underline{Case 1}} && \text{\underline{Case 2}}\\3x+4 & < -6 \quad & 3x+4 & > 6\\3x & < -10 \quad & 3x & > 2\\x & < -(10)/(3) \quad & x & > (2)/(3)\end{aligned}`

So the solution to the inequality in interval notation is:

`\left(- \infty, -(10)/(3)\right) \cup \left((2)/(3), \infty\right)`