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 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 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 \(x = 3\) (since \(x = 3\) is not included in the solution set) and an arrow pointing to the right (towards positive infinity). Looking at the options, the second number line (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, i.e., the option with \(-5\) \(-4\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\circ\) \(3\) \(4\) \(5\) and the arrow to the right)