Given: The equation and inequalty below
[tex]\begin{gathered} y=|3x-12|+1 \\ y<5 \end{gathered}[/tex]
To Determine: The values of x satisfying the given conditions using interval notation
Solve the first equation
[tex]y=|3x-12|+1[/tex][tex]\mathrm{Domain\: of\: }\: \mleft|3x-12\mright|+1\: \colon\quad \begin{bmatrix}\mathrm{Solution\colon}\: & \: -\infty\: The range[tex]\mathrm{Range\: of\: }\mleft|3x-12\mright|+1\colon\quad \begin{bmatrix}\mathrm{Solution\colon}\: & \: f\mleft(x\mright)\ge\: 1\: \\ \: \mathrm{Interval\: Notation\colon} & \: \lbrack1,\: \infty\: )\end{bmatrix}[/tex]
The y-intercept, make x = 0
[tex]\begin{gathered} y=\mathrm{\: }\mleft|3x-12\mright|+1 \\ y=|3(0)-12|+1 \\ y=|0-12|+1 \\ y=|-12|+1 \\ y=12+1=13 \\ T_{he\text{ coordinate of the y intercept is}} \\ (0,13) \end{gathered}[/tex]
The minimum point
[tex]\begin{gathered} T_{he\text{ x coordinate of the minimum point}} \\ 3x-12=0 \\ 3x=12 \\ x=\frac{12}{3}=4 \\ T_{he\text{ y cordinate of the minimum point}} \\ y=|3x-12|+1 \\ y=|3(4)-12|+1 \\ y=|12-12|+1 \\ y=1 \\ T_{he\text{ coordinate of the minimum point is}}=(4,1) \end{gathered}[/tex]
Let us graph the two equation as shown below
From the graph above, the set of values of x that satisfies the equation and inequality can be seen from point A to point B.
Hence,
The solution is 2.667 < x < 5.333
Using interval notation we have (2.667, 5.333)
