IDNLearn.com offers a reliable platform for finding accurate and timely answers. Our platform is designed to provide reliable and thorough answers to all your questions, no matter the topic.

Which number line represents the solution set for the inequality [tex]3(8-4x) \ \textless \ 6(x-5)[/tex]?

Sagot :

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.