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

Question

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the left.
A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.
A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3 and a bold line starts at negative 3 and is pointing to the left.
A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3 and a bold line starts at negative 3 and is pointing to the right.

Answer 1

To solve the inequality \(3(8 - 4x) < 6(x - 5)\), let's start by simplifying both sides of the inequality.

1. Distribute on both sides: \[ 24 - 12x < 6x - 30 \]

2. Now, let's add \(12x\) to both sides: \[ 24 < 18x - 30 \]

3. Next, add \(30\) to both sides: \[ 54 < 18x \]

4. Now, divide both sides by \(18\): \[ 3 < x \] or, equivalently, \[ x > 3 \]

Now, we need to find the correct representation of this solution set on a number line. The solution \(x > 3\) indicates that:

• There should be an open circle at \(3\) (indicating that \(3\) is not included in the solution).
• The bold line should start at \(3\) and point to the right, indicating that all numbers greater than \(3\) are included in the solution.

Therefore, the correct answer is:

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.