Problem 28 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ 0<\frac{7 x-5}{3} \leq 4 $$

Step-by-Step Solution

Answer

The solution is \(\left(\frac{5}{7}, \frac{17}{7}\right]\).

Step 1: Understand the Inequality Setup

The inequality is given as two conditions: \(0 < \frac{7x - 5}{3}\) and \(\frac{7x - 5}{3} \leq 4\). We need to solve both conditions separately and then combine the solutions.

Step 2: Solve the First Inequality

Solve \(0 < \frac{7x - 5}{3}\). Multiply both sides by 3 to eliminate the fraction:\[0 < 7x - 5\]Add 5 to both sides:\[5 < 7x\]Divide both sides by 7:\[\frac{5}{7} < x\].

Step 3: Solve the Second Inequality

Solve \(\frac{7x - 5}{3} \leq 4\). Multiply both sides by 3 to eliminate the fraction:\[7x - 5 \leq 12\]Add 5 to both sides:\[7x \leq 17\]Divide both sides by 7:\[x \leq \frac{17}{7}\].

Step 4: Combine the Solutions

Now combine the solutions from both inequalities:\[\frac{5}{7} < x \leq \frac{17}{7}\]The solution can be expressed in interval notation as:\(\left(\frac{5}{7}, \frac{17}{7}\right]\).

Key Concepts

Interval Notation

Interval notation is a way of describing a set of numbers along a number line. It is compact and clearly shows the start and end of the range, as well as which endpoints are included and which are excluded. In the interval notation

• parentheses "(" or ")" indicate that the endpoint is not included, known as open.
• square brackets "[" or "]" indicate the endpoint is included, known as closed.

In this inequality solution, we found that \[\frac{5}{7} < x \leq \frac{17}{7}\]This result is expressed in interval notation as:\(\left(\frac{5}{7}, \frac{17}{7}\right]\)This tells us "x" can be any number greater than \(\frac{5}{7}\) but less than or equal to \(\frac{17}{7}\). The parentheses around \(\frac{5}{7}\) show it is not included, while the bracket around \(\frac{17}{7}\) indicates it is included.

Set-Builder Notation

Set-builder notation offers a flexible way to represent sets by describing the properties that its members must satisfy. It it usually formatted like this:

• {variable | statement about the variable}

For instance, if we need to describe the set of numbers we found from solving the inequality,\[\frac{5}{7} < x \leq \frac{17}{7}\]in set-builder notation, it would look like:\(\{x | \frac{5}{7} < x \leq \frac{17}{7} \}\)This reads as "the set of all \(x\) such that \(x\) is greater than \(\frac{5}{7}\) and less than or equal to \(\frac{17}{7}\)." Set-builder notation is particularly helpful when you need to specify more detailed conditions or constraints that are harder to capture with interval notation alone.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations or inequalities to isolate the variable of interest. This can include operations like addition, subtraction, multiplication, division, and distribution. Here's a step-by-step illustration of how we solved the inequality.
Initially, we had \[0 < \frac{7x - 5}{3}\]Multiply by 3 to remove the fraction:\[0 < 7x - 5\]Then, add 5 to both sides to isolate the term with "x":\[5 < 7x\]Finally, divide everything by 7:\[\frac{5}{7} < x\]
We repeated a similar process for the second condition \[\frac{7x - 5}{3} \leq 4\]Starting by multiplying by 3,\[7x - 5 \leq 12\]Adding 5:\[7x \leq 17\]And dividing by 7 gives:\[x \leq \frac{17}{7}\]
Algebraic manipulation is a fundamental skill that helps move terms around effectively to find the range of solutions. Each step in algebraic manipulation should maintain the equality or inequality sign to preserve correct relationships between expressions.