Given the inequality:
[tex]2-|3x-5|>-7[/tex]
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.
[tex]\begin{gathered} 2-|3x-5|-2>-7-2 \\ 2-2-|3x-5|>-9 \\ -|3x-5|>-9 \end{gathered}[/tex]
Now, multiply the equation by -1:
[tex]|3x-5|<9[/tex]
If |3x - 5| < 9, then 3x - 5 < 9 or -(3x - 5) < 9.
Step 02: Find the interval in which 3x - 5 < 9.
[tex]3x-5<9[/tex]
Isolate x by adding 5 to both sides. In sequence, divide the sides by 3:
[tex]\begin{gathered} 3x-5+5<9+5 \\ 3x<14 \\ \frac{3x}{3}<\frac{14}{3} \\ x<\frac{14}{3} \end{gathered}[/tex]
Step 03: Find the interval in which -(3x - 5) < 9.
[tex]-3x+5<9[/tex]
To isolate x, first, subtract 5 from both sides. Second, divide the sides by 3. Finally, multiply the inequality by -1.
[tex]\begin{gathered} -3x+5-5<9-5 \\ -3x<4 \\ \frac{-3x}{3}<\frac{4}{3} \\ -x<\frac{4}{3} \\ x>-\frac{4}{3} \end{gathered}[/tex]
Step 04: Graph the interval.
Since x < 14/3 and x > -4/3:
Graphing the answer:
Step 05: Write your answer in interval notation.
[tex]-\frac{4}{3}In interval notation:
[tex](-\frac{4}{3},\frac{14}{3})[/tex]
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).
[tex]\begin{gathered} 2-|3x-5|>-7 \\ \end{gathered}[/tex]
Substituting x by 0.
[tex]\begin{gathered} 2-|3\cdot0-5|>-7 \\ 2-|0-5|>-7 \\ 2-|5|>-7 \\ 2-5>-7 \\ -3>-7 \end{gathered}[/tex]
True.
Substituting x by 5.
[tex]\begin{gathered} 2-|3x-5|>-7 \\ 2-|3\cdot5-5|>-7 \\ 2-|15-5|>-7 \\ 2-|10|>-7 \\ -8>-7 \end{gathered}[/tex]
False, since -8 is not greater than -7. It is expected since 5 is not an answer for this inequality.
Answer:
[tex](-\frac{4}{3},\frac{14}{3})[/tex]
