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Chapter 1: Problem 55

Graph all solutions on a number line and provide the corresponding interval notation. $$ 8 x-3 \leq 1 \text { or } 6 x-7 \geq 8 $$

Short Answer

Expert verified
\((-\infty, \frac{1}{2}] \cup [\frac{5}{2}, \infty)\)

Step by step solution

01

Solve the first inequality

Start by isolating the variable in the inequality \(8x - 3 \leq 1\). Add 3 to both sides to get: \(8x \leq 4\). Divide each side by 8 to solve for \(x\), resulting in \(x \leq \frac{1}{2}\).
02

Solve the second inequality

Now solve \(6x - 7 \geq 8\). Add 7 to both sides to obtain \(6x \geq 15\). Divide each side by 6 to find \(x \geq \frac{5}{2}\).
03

Graph the solution

For both inequalities, graph \(x \leq \frac{1}{2}\) and \(x \geq \frac{5}{2}\) on a number line. Shade the region left of \(\frac{1}{2}\) and the region right of \(\frac{5}{2}\), including the points \(\frac{1}{2}\) and \(\frac{5}{2}\) as both are inclusive (closed circles).
04

Write the interval notation

Since we are dealing with 'or', the solution set is the union of the two intervals. For \(x \leq \frac{1}{2}\), the interval is \((-\infty, \frac{1}{2}]\). For \(x \geq \frac{5}{2}\), the interval is \([\frac{5}{2}, \infty)\). Therefore, the solution in interval notation is: \((-\infty, \frac{1}{2}] \cup [\frac{5}{2}, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Number Line
The number line is a visual representation of numbers laid out in a straight path. It's incredibly useful when dealing with inequalities, as it helps to see the range of possible values that satisfy a condition.
To graph inequalities like \(x \leq \frac{1}{2}\) and \(x \geq \frac{5}{2}\), we take specific steps:
  • Identify the key points, such as \(\frac{1}{2}\) and \(\frac{5}{2}\).
  • Use closed circles when the inequality includes \(\leq\) or \(\geq\), meaning the endpoint is part of the solution.
  • Shade to the left for \(x \leq \frac{1}{2}\), indicating all numbers less than or equal to \(\frac{1}{2}\) are solutions.
  • Shade to the right for \(x \geq \frac{5}{2}\), showing that all numbers greater than or equal to \(\frac{5}{2}\) are included.
This shading technique helps visualize the solution set, making it easier to understand both inequalities on the same line.
Interval Notation Demystified
Interval notation is a compact way to express a range of values. It's especially handy in mathematics for indicating solutions to inequalities.
To convert our number line graph to interval notation, follow these guidelines:
  • Start with the smallest number in the interval and move to the largest.
  • Use square brackets \([ ]\) for numbers that are included in the interval, indicated by \(\leq\) or \(\geq\).
  • Use parentheses \(( )\) to show that a number is not included, such as \(infin\) or when \(<\) or \(>\) are involved.
  • "\(( -\infty, \frac{1}{2}] \)" means all numbers from negative infinity up to and including \(\frac{1}{2}\).
  • "\([\frac{5}{2}, \infty)\)" encompasses all numbers from \(\frac{5}{2}\) to positive infinity.
Interval notation provides a precise way to capture the solution set for an inequality, making these intervals clear and concise.
Exploring the Union of Intervals
When solving inequalities like \(8x - 3 \leq 1\) or \(6x - 7 \geq 8\), the term 'or' indicates a union of solutions.
The union of intervals combines multiple solution sets:
  • For "or" situations, any number satisfying either inequality belongs to the solution.
  • In our case, we combine the intervals from \(( -\infty, \frac{1}{2}]\) and \([\frac{5}{2}, \infty)\) using the union symbol \(\cup\).
  • This is shown as \(( -\infty, \frac{1}{2}] \cup [\frac{5}{2}, \infty)\), which reads as one unified set.
The union effectively captures every part of the number line that satisfies at least one inequality, encompassing the entire solution range seamlessly.

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