Questions: Solve the following inequality. Write the solution set in interval notation. 3(x-4)<2(2x-1)

Solution Steps

To solve the inequality \(3(x-4) < 2(2x-1)\), we need to follow these steps:

1. Distribute the constants on both sides of the inequality.
2. Combine like terms to isolate the variable \(x\) on one side.
3. Solve for \(x\).
4. Write the solution set in interval notation.

First, we need to distribute the constants on both sides of the inequality: \[ 3(x - 4) < 2(2x - 1) \]

Distribute the 3 on the left side: \[ 3x - 12 \]

Distribute the 2 on the right side: \[ 4x - 2 \]

So the inequality becomes: \[ 3x - 12 < 4x - 2 \]

Next, we need to move all terms involving \(x\) to one side of the inequality. We can do this by subtracting \(3x\) from both sides: \[ 3x - 12 - 3x < 4x - 2 - 3x \]

Simplify: \[ -12 < x - 2 \]

Now, we need to isolate \(x\) by adding 2 to both sides of the inequality: \[ -12 + 2 < x - 2 + 2 \]

Simplify: \[ -10 < x \]

The inequality \(-10 < x\) can be written in interval notation as: \[ (-10, \infty) \]

Final Answer