To solve the inequality [tex]\(3(8 - 4x) < 6(x - 5)\)[/tex], follow these steps:
1. Distribute coefficients on both sides:
[tex]\[3 \cdot 8 - 3 \cdot 4x < 6 \cdot x - 6 \cdot 5\][/tex]
This simplifies to:
[tex]\[24 - 12x < 6x - 30\][/tex]
2. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other side:
Add [tex]\(12x\)[/tex] to both sides:
[tex]\[24 < 18x - 30\][/tex]
Add 30 to both sides:
[tex]\[54 < 18x\][/tex]
3. Solve for [tex]\(x\)[/tex] by dividing both sides by 18:
[tex]\[\frac{54}{18} < x\][/tex]
[tex]\[3 < x\][/tex]
which can also be written as:
[tex]\[x > 3\][/tex]
The solution set for the inequality [tex]\(3(8 - 4x) < 6(x - 5)\)[/tex] is [tex]\(x > 3\)[/tex].
4. Representing this on the number line:
- Draw a number line.
- Mark the point [tex]\(3\)[/tex] with an open circle to indicate that [tex]\(3\)[/tex] is not included in the solution ([tex]\(x\)[/tex] is strictly greater than [tex]\(3\)[/tex]).
- Shade the number line to the right of [tex]\(3\)[/tex] to indicate all the numbers greater than [tex]\(3\)[/tex].
Here is the visual representation of the solution set on the number line:
[tex]\[
\begin{array}{ccccccccccccccc}
&&&&&&&& 3 && 4 && 5 && 6 & \cdots\\
\hline
&- & - & - & - & - & - & - &\circ &===== &===== &===== &===== &===== &===== & \to \\
&&&&&&&& \quad \text{ } &&&&&&&& \quad x
\end{array}
\][/tex]
This number line represents the solution [tex]\(x > 3\)[/tex], where the shaded area starts just after 3 and continues indefinitely to the right.
