Question

PLEASE HELP
Solve the inequality. | 2x + 3 | < 7

please show all the work

Answer 1

`-5<x<2`

Explanation:

We have the inequality:

`|2x+3|<7`

Use the definition of absolute value:

`(2x+3)<7\text{ or } -(2x+3)<7`

Let's evaluate each case individually.

Case 1)

We have:

`2x+3<7`

Subtract 3 from both sides:

`2x<4`

Divide both sides by 2:

`x<2\\`

Case 2)

We have:

`-(2x+3)<7`

Divide both sides by -1. Since we're dividing by a negative, flip our sign:

`2x+3<-7`

Subtract 3 from both sides:

`2x<-10`

Divide both sides by 2:

`x<-5`

So, our solutions are, from least to greatest:

`\-5, 2\`

We can see that our inequality is a greater than.

Remember, when dealing with absolute value:

When our sign is greater than (or equal to), our solution is an "or" inequality with our solutions being all values less than our lesser solution or all values greater than our greater solution.

When our sign is less than (or equal to), our solution is an "and" inequality with our solution being between our two solutions.

Since our inequality is a less than, our solution is all the values between our lesser solution and our greater solution.

Our lesser solution is -5. Our greater solution is 2.

So, the solution to the inequality is all values between -5 and 2. As a compound inequality, this is:

`-5<x<2`

And we're done!