Question

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

Explanation:

Step1: Expand both sides

First, we expand the left - hand side and the right - hand side of the inequality \(3(8 - 4x)<6(x - 5)\).
Using the distributive property \(a(b + c)=ab+ac\), for the left - hand side: \(3\times8-3\times4x = 24-12x\).
For the 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

We add \(12x\) to both sides of the inequality to get all the \(x\) terms on one side.
\(24-12x + 12x<6x-30 + 12x\), which simplifies to \(24<18x-30\).

Step3: Move the constant terms to the other side

We add 30 to both sides of the inequality: \(24 + 30<18x-30 + 30\).
Calculating the left - hand side: \(54<18x\).

Step4: Solve for x

We 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 we want all values greater than 3).

Answer:

The second number line (the one with an open circle at 3 and the blue arrow pointing to the right, passing through 4, 5, etc.)