If 8x -3 <0 and 8x>0, express these in a continued inequality and find its solution

If 8x -3 <0 and 8x>0, express these in a continued inequality and find its solution.

The correct answer and explanation is:

Given:

We have two inequalities:

1. 8x−3<08x – 3 < 0
2. 8x>08x > 0

We need to combine them into a compound (continued) inequality and then solve it.

Step 1: Solve each inequality individually.

Inequality 1:

8x−3<08x – 3 < 0

Add 3 to both sides: 8x<38x < 3

Divide both sides by 8: x<38x < \frac{3}{8}

Inequality 2:

8x>08x > 0

Divide both sides by 8: x>0x > 0

Step 2: Combine into a continued inequality.

From above: 0<x<380 < x < \frac{3}{8}

✅ Final Answer:

0<x<38\boxed{0 < x < \frac{3}{8}}

📘 Explanation (Approx. 300 words):

In this problem, we are given two separate inequalities: 8x−3<08x – 3 < 0 and 8x>08x > 0. The goal is to combine them into a single continued inequality and find the range of values for xx that satisfy both conditions simultaneously.

Let’s first solve each inequality one at a time. Starting with 8x−3<08x – 3 < 0, we isolate xx by first adding 3 to both sides, giving 8x<38x < 3, and then divide by 8 to get x<38x < \frac{3}{8}. This means that xx must be less than 38\frac{3}{8}.

Now, let’s solve the second inequality 8x>08x > 0. Divide both sides by 8 to get x>0x > 0. This tells us that xx must also be greater than 0.

To find the values of xx that satisfy both inequalities, we look for the overlap or intersection between the two solutions. x<38x < \frac{3}{8} overlaps with x>0x > 0 in the range where: 0<x<380 < x < \frac{3}{8}

This is called a compound inequality, which shows a range of numbers that satisfy two or more conditions. Visually, on a number line, this would be an open interval between 0 and 38\frac{3}{8}, excluding both endpoints.

In conclusion, the solution to the combined inequality is: 0<x<38\boxed{0 < x < \frac{3}{8}}

This means xx can be any real number between 0 and 38\frac{3}{8}, but not equal to either endpoint.