Problem 43 - Intermediate Algebra

Question

Solve each compound inequality. Graph the solution set and write it using interval notation. \(2 x>x+3\) or \(\frac{x}{8}+1<\frac{13}{8}\)

Step-by-Step Solution

Answer

Interval notation: \((-\infty, 5) \cup (3, \infty)\).

Step 1: Solve the First Inequality

We start by solving the inequality \( 2x > x + 3 \). Subtract \( x \) from both sides of the inequality to get \( 2x - x > 3 \). Simplifying, we have \( x > 3 \). This is our first solution.

Step 2: Solve the Second Inequality

Next, solve the inequality \( \frac{x}{8} + 1 < \frac{13}{8} \). Subtract 1 from both sides to isolate the variable term: \( \frac{x}{8} < \frac{13}{8} - \frac{8}{8}\). Simplifying the right side, we have \( \frac{x}{8} < \frac{5}{8} \). Multiply both sides by 8 to clear the fraction: \( x < 5 \). This is our second solution.

Step 3: Combine the Solutions

The compound inequality is \( x > 3 \) or \( x < 5 \). This means the solution consists of two intervals: one for \( x > 3 \) and another for \( x < 5 \). Since this is an 'or' statement, the solution is from both inequalities.

Step 4: Write in Interval Notation and Graph the Solution

In interval notation, the solution is \( (-\infty, 5) \cup (3, \infty) \). To graph the solution set, use a number line: graph an open interval starting just beyond 3 to the right indefinitely, and graph an open interval starting just before 5 to the left indefinitely. This shows that any number less than 5 or greater than 3 satisfies the compound inequality.

Key Concepts

Interval Notation

Interval notation is a shorthand way of writing the set of all numbers between two endpoints. It uses parentheses and brackets to describe open and closed intervals. There are a few key symbols and terms to remember:

• Parentheses \((\text{ or } )\): These are used for open intervals, which do not include the endpoint values. For example, \((a, b)\) means all the numbers between \(a\) and \(b\), but not including \(a\) or \(b\).
• Brackets \([\text{ or } ]\): These indicate closed intervals, which do include the endpoints. For example, \([a, b]\) includes \(a\) and \(b\) along with all the numbers in between.
• Infinity \((\infty)\): Since infinity isn't a number, we always use parentheses to show that intervals extending to infinity do not include a boundary.

In the solution to the exercise, we have \((-\infty, 5)\) and \((3, \infty)\), meaning the interval goes up to but does not include 5 and starts from just after 3.

Inequality Solving

Inequality solving involves finding the range of values for a variable that makes the inequality true. It's like solving regular equations, but with special attention to the inequality sign.

• First, you treat each part of a compound inequality separately, simplifying each inequality step by step.
• The inequality symbols tell us the range of possible solutions:
  • ">" and "<": Greater than and less than, indicating solutions that exclude the value.
  • "≥" and "≤": Greater than or equal to, and less than or equal to, indicating solutions that include the value.
• Apply basic algebraic operations like adding, subtracting, multiplying, or dividing both sides while remembering to flip the inequality sign when multiplying or dividing by a negative number.

In the exercise, the inequalities are solved separately, resulting in the solutions \(x > 3\) and \(x < 5\). These are then combined due to the 'or' condition.

Graphing Solutions

Graphing solutions of inequalities helps visualize the range of possible solutions on a number line. Here's how you can represent it:

• Draw a baseline representing the number line.
• An open circle is used to denote numbers not inclusive in the solution, while a filled circle represents numbers that are inclusive.
• For the solution \(x > 3\), place an open circle at 3 and shade all numbers to the right. For \(x < 5\), place an open circle at 5 and shade all numbers to the left.
• If two graphs overlap because of an 'and' statement, you shade only the overlap. However, in an 'or' statement like this exercise, you shade both directions starting from their respective points.

By graphing on a number line, you can visually verify which numbers satisfy either part of the compound inequality. This exercise’s graph denotes that any number less than 5 or greater than 3 is valid, leaving out the numbers between 3 and 5.