Question

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

Short answer

The solution of the inequality is \(0 < x < \frac{3}{8}\).

Step-by-step solution

Understand the Given Inequalities

We have two inequalities given: 1) \(8x - 3 < 0\) 2) \(8x > 0\)These inequalities tell us different things about the possible values of \(x\). The goal is to solve them individually first before combining them into a continued inequality.

Solve the First Inequality

To solve \(8x - 3 < 0\), we first add 3 to both sides:\(8x - 3 + 3 < 0 + 3\)Simplifying gives:\(8x < 3\)Next, divide both sides by 8:\(x < \frac{3}{8}\)

Solve the Second Inequality

To solve \(8x > 0\), divide both sides of the inequality by 8:\(x > \frac{0}{8}\)Simplifying gives:\(x > 0\)

Combine the Inequalities

Now we combine \(x < \frac{3}{8}\) and \(x > 0\) into a single continued inequality. By combining, we get:\(0 < x < \frac{3}{8}\).This states that \(x\) is greater than 0 but less than \(\frac{3}{8}\).

Solution Verification

Verify each part of the combined inequality:- For \(x > 0\): If \(x = 0.1\), then \(8x = 0.8 > 0\).- For \(x < \frac{3}{8}\): If \(x = 0.3\), then \(8x = 2.4\) and \(8x - 3 = 2.4 - 3 = -0.6 < 0\).Both conditions hold true within the range found.

Key concepts

Inequality Solutions

Solving inequalities is much like solving equations, but with a focus on the relationship between expressions rather than their equality. In our exercise, two inequalities were present:

• \(8x - 3 < 0\)
• \(8x > 0\)

These inequalities restricted the values that \(x\) could hold. The process began by solving each one independently. For inequality solutions, we treat them similarly to equations, performing operations such as addition, subtraction, multiplication, or division on both sides. However, if we multiply or divide by a negative number, we must flip the inequality sign. Through systematic steps, we arrived at two conditions, \(x < \frac{3}{8}\) and \(x > 0\). The solutions reveal ranges of values instead of specific numbers.

Continued Inequalities

Once individual inequalities are solved, they can be combined into a continued inequality. This captures the shared range of possible values that satisfy all conditions simultaneously. In our case, we combined:

• \(x < \frac{3}{8}\)
• \(x > 0\)

into a single statement: \(0 < x < \frac{3}{8}\).

Essentially, a continued inequality strings together different inequalities to provide one clear picture of the permissible range for the variable. This can be thought of as expressing several conditions that overlap, portrayed as a straightforward set of boundaries for \(x\). The continued inequality neatly summarizes where all the original individual inequalities align.

Mathematical Problem-Solving

Solving mathematical problems often involves a combination of strategic thinking and systematic steps. With inequalities, we must carefully handle each operation and understand the rules that apply especially to inequalities.

When tackling these problems:

• Break down complex problems into simpler parts.
• Solve each part step-by-step.
• Look to combine results where applicable.

Remember, double-check your operations, especially when dividing or multiplying by any number. Incorrectly handling an inequality can lead to a wrong solution. Always verify your solutions by plugging them back into the original inequalities. This ensures that all criteria are satisfied and your problem-solving process is valid.

Combining Inequalities

Combining inequalities is an essential skill in mathematics that allows us to simplify multiple conditions into a single expression. This often involves aligning the inequalities along the same variable. For instance, from our solution:

• We derived \(x < \frac{3}{8}\) and \(x > 0\).
• Combined, this yields \(0 < x < \frac{3}{8}\).

Combining inequalities values the power of intersection in sets — the combined inequality identifies a range where all conditions coexist. This method greatly simplifies the interpretation of inequality systems and helps give a comprehensive look at the valid solutions which, in the context of this exercise, is any \(x\) greater than \(0\) but less than \(\frac{3}{8}\). By mastering this technique, one develops a deeper understanding of how inequality constraints interact in mathematics.