Question

which number line represents the solution set for the inequality 3(8 − 4x) < 6(x − 5)? four number line options: first with blue left of 3, second blue right of 3, third blue left of -3, fourth blue right of -3, each with open circles at 3 or -3 and arrows in respective directions

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 \(a(b + c)=ab+ac\), 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

We 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

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

Step4: Solve for x

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

On a number line, the solution \(x > 3\) is represented by an open circle at \(x = 3\) (since \(x\) is not equal to 3) and an arrow pointing to the right (since \(x\) is greater than 3). Looking at the given number lines, the second number line (the one with an open circle at 3 and the blue line to the right of 3) represents the solution set.

Answer:

The second number line (the one with an open circle at 3 and the blue line extending to the right from 3)