Answer :
Answer:
Solutions: [tex]\sf x = 0[/tex], [tex]\sf x = \dfrac{1}{4}[/tex], [tex]\sf x = \dfrac{1}{2}[/tex]
Non-Solutions: [tex]\sf x = -1[/tex], [tex]\sf x = \dfrac{2}{3}[/tex]
Step-by-step explanation:
To solve the inequality [tex]\sf |12x - 1| < 7[/tex], we need to consider the two cases that arise from the absolute value inequality:
[tex] \begin{cases} \sf 12x - 1 < 7\\12x - 1 > -7\end{cases}[/tex]
Let's solve each case to find the solutions and non-solutions.
Case 1:
[tex]\sf 12x - 1 < 7[/tex]
[tex]\sf 12x - 1 < 7 [/tex]
Add 1 to both sides:
[tex]\sf 12x < 8 [/tex]
Divide both sides by 12:
[tex]\sf x < \dfrac{8}{12} [/tex]
[tex]\sf x < \dfrac{2}{3} [/tex]
Case 2:
[tex]\sf 12x - 1 > -7[/tex]
[tex]\sf 12x - 1 > -7 [/tex]
Add 1 to both sides:
[tex]\sf 12x > -6 [/tex]
Divide both sides by 12:
[tex]\sf x > \dfrac{-6}{12} [/tex]
[tex]\sf x > -\dfrac{1}{2} [/tex]
Now, combining the results from both cases, we have:
[tex]\sf -\dfrac{1}{2} < x < \dfrac{2}{3} [/tex]
This interval represents all values of [tex]\sf x[/tex] that satisfy [tex]\sf |12x - 1| < 7[/tex].
Identifying Solutions and Non-Solutions:
Solution 1:
[tex]\sf x = 0[/tex]
Substitute [tex]\sf x = 0[/tex] into the original inequality:
[tex]\sf |12(0) - 1| = |-1| = 1 < 7 [/tex]
Therefore, [tex]\sf x = 0[/tex] is a solution.
Solution 2:
[tex]\sf x = \dfrac{1}{4}[/tex]
Substitute [tex]\sf x = \dfrac{1}{4}[/tex] into the original inequality:
[tex]\sf |12\left(\dfrac{1}{4}\right) - 1| = |3 - 1| = 2 < 7 [/tex]
Therefore, [tex]\sf x = \dfrac{1}{4}[/tex] is a solution.
Solution 3:
[tex]\sf x = \dfrac{1}{2}[/tex]
Substitute [tex]\sf x = \dfrac{1}{2}[/tex] into the original inequality:
[tex]\sf |12\left(\dfrac{1}{2}\right) - 1| = |6 - 1| = 5 < 7 [/tex]
Therefore, [tex]\sf x = \dfrac{1}{2}[/tex] is a solution.
Non-Solution 1:
[tex]\sf x = -1[/tex]
Substitute [tex]\sf x = -1[/tex] into the original inequality:
[tex]\sf |12(-1) - 1| = |-12 - 1| = |-13| = 13 \not< 7 [/tex]
Therefore, [tex]\sf x = -1[/tex] is not a solution.
Non-Solution 2:
[tex]\sf x = \dfrac{2}{3}[/tex]
Substitute [tex]\sf x = \dfrac{2}{3}[/tex] into the original inequality:
[tex]\sf |12\left(\dfrac{2}{3}\right) - 1| = |8 - 1| = 7 \not< 7 [/tex]
Therefore, [tex]\sf x = \dfrac{2}{3}[/tex] is not a solution.
Summary:
Solutions: [tex]\sf x = 0[/tex], [tex]\sf x = \dfrac{1}{4}[/tex], [tex]\sf x = \dfrac{1}{2}[/tex]
Non-Solutions: [tex]\sf x = -1[/tex], [tex]\sf x = \dfrac{2}{3}[/tex]
The solutions are values of [tex]\sf x[/tex] that satisfy [tex]\sf |12x - 1| < 7[/tex], while the non-solutions are values of [tex]\sf x[/tex] that do not satisfy the inequality.
