Question

which number line represents the solution set for the inequality 3(8 − 4x) < 6(x − 5)?

Explanation:

Step1: Expand both sides

First, expand the left - hand side and the right - hand side of the inequality \(3(8 - 4x)<6(x - 5)\).
Using the distributive property \(a(b + c)=ab+ac\), we have:
Left - hand side: \(3\times8-3\times4x = 24-12x\)
Right - hand side: \(6\times x-6\times5=6x - 30\)
So the inequality becomes \(24-12x<6x - 30\).

Step2: Move the terms with x to one side

Add \(12x\) to both sides of the inequality to get all the \(x\) terms on one side.
\(24-12x + 12x<6x-30 + 12x\)
Simplifying, we have \(24<18x-30\).

Step3: Move the constant terms to the other side

Add 30 to both sides of the inequality:
\(24 + 30<18x-30 + 30\)
\(54<18x\)

Step4: Solve for x

Divide both sides of the inequality by 18:
\(\frac{54}{18}<\frac{18x}{18}\)
Simplifying, we get \(3 < x\) or \(x>3\).

On a number line, the solution \(x > 3\) is represented by an open circle at \(x = 3\) (because the inequality is strict, \(x
eq3\)) and an arrow pointing to the right (since \(x\) is greater than 3).

Looking at the options, the second number line (the one with an open circle at 3 and the arrow pointing to the right) represents the solution set.

Answer:

The second number line (the one with an open circle at 3 and the arrow pointing to the right among the given options)