Question

Need help with this. This is all one problem. The directions are numbered 1-6

Answer 1

Given the inequality:

`2-|3x-5|>-7`

To find x, follow the steps below.

Step 01: Isolate the absolute value.

To do it, first, subtract 2 from both sides of the inequality.

`\begin{gathered} 2-|3x-5|-2>-7-2 \\ 2-2-|3x-5|>-9 \\ -|3x-5|>-9 \end{gathered}`

Now, multiply the equation by -1:

`|3x-5|<9`

If |3x - 5| < 9, then 3x - 5 < 9 or -(3x - 5) < 9.

Step 02: Find the interval in which 3x - 5 < 9.

`3x-5<9`

Isolate x by adding 5 to both sides. In sequence, divide the sides by 3:

`\begin{gathered} 3x-5+5<9+5 \\ 3x<14 \\ (3x)/(3)<(14)/(3) \\ x<(14)/(3) \end{gathered}`

Step 03: Find the interval in which -(3x - 5) < 9.

`-3x+5<9`

To isolate x, first, subtract 5 from both sides. Second, divide the sides by 3. Finally, multiply the inequality by -1.

`\begin{gathered} -3x+5-5<9-5 \\ -3x<4 \\ (-3x)/(3)<(4)/(3) \\ -x<(4)/(3) \\ x>-(4)/(3) \end{gathered}`

Step 04: Graph the interval.

Since x < 14/3 and x > -4/3:

Graphing the answer:

Step 05: Write your answer in interval notation.

`-(4)/(3)In interval notation: <p></p>[tex](-(4)/(3),(14)/(3))`

Step 06: Double-check the solution.

To double-check the solution, choose one point inside the interval and observe if the answer fits the equation. You can also choose a point outside the interval.

Let's choose x = 0 (inside the interval) and x = 5 (outside the interval).

`\begin{gathered} 2-|3x-5|>-7 \\ \end{gathered}`

Substituting x by 0.

`\begin{gathered} 2-|3\cdot0-5|>-7 \\ 2-|0-5|>-7 \\ 2-|5|>-7 \\ 2-5>-7 \\ -3>-7 \end{gathered}`

True.

Substituting x by 5.

`\begin{gathered} 2-|3x-5|>-7 \\ 2-|3\cdot5-5|>-7 \\ 2-|15-5|>-7 \\ 2-|10|>-7 \\ -8>-7 \end{gathered}`

False, since -8 is not greater than -7. It is expected since 5 is not an answer for this inequality.

Answer:

`(-(4)/(3),(14)/(3))`