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 rid of the \(- 12x\) on the left.
\(24-12x + 12x<6x-30 + 12x\), which simplifies to \(24<18x-30\).

Step3: Isolate the constant term

We add \(30\) to both sides of the inequality.
\(24 + 30<18x-30 + 30\), so \(54<18x\).

Step4: Solve for x

We divide both sides of the inequality by \(18\).
\(\frac{54}{18}<\frac{18x}{18}\), which gives \(3 < x\) or \(x>3\).

On a number line, the solution \(x > 3\) is represented by an open circle at \(3\) (since \(x\) is not equal to \(3\)) and an arrow pointing to the right (towards positive infinity).

Answer:

The number line with an open circle at \(3\) and an arrow pointing to the right (the one where the shaded region is to the right of \(3\) with an open circle at \(3\)) represents the solution set.