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Chapter 2: Problem 71

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ -\frac{2}{3} \geq \frac{2 y}{3}-\frac{3}{4} $$

Short Answer

Expert verified
The solution is \((-\infty, \frac{1}{8} ]\).

Step by step solution

01

Eliminate Fractions

To eliminate fractions, we will multiply every term by the least common multiple of the denominators. Here, the denominators are 3 and 4, so the least common multiple is 12. Multiply everything by 12.\(12 \left(-\frac{2}{3}\right) \geq 12 \left(\frac{2y}{3}\right) - 12 \left(\frac{3}{4}\right) \)This simplifies to:\(-8 \geq 8y - 9\)
02

Isolate the Variable Term

First, add 9 to both sides of the inequality to move the constant term on the right to the left.\(-8 + 9 \geq 8y - 9 + 9\)Simplify:\(1 \geq 8y\)This can also be written as:\(8y \leq 1\)
03

Solve for the Variable

To isolate \(y\) on one side, divide both sides by 8.\(\frac{8y}{8} \leq \frac{1}{8}\)Simplify:\(y \leq \frac{1}{8}\)
04

Write the Solution in Interval Notation

The solution \(y \leq \frac{1}{8}\) means that \(y\) can be any number less than or equal to \(\frac{1}{8}\). In interval notation, this is written as:\(( -\infty, \frac{1}{8} ]\)
05

Graph the Solution

To graph the solution \(( -\infty, \frac{1}{8} ]\), draw a number line. Locate \(\frac{1}{8}\) on the number line. Use a closed dot on \(\frac{1}{8}\) to indicate that it is included in the solution set, and shade all the area to the left of \(\frac{1}{8}\) to represent all values less than \(\frac{1}{8}\).

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Key Concepts

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

Interval Notation
Interval notation is a way to represent a set of numbers between two endpoints. It's particularly useful when dealing with inequalities as it provides a clear and concise way to display solution sets.
For example, consider a solution set of all numbers less than or equal to \(\frac{1}{8}\). We would write this in interval notation as \(( -\infty, \frac{1}{8} ]\).
This notation includes:
  • A round bracket "(" or ")" implies that the endpoint is not included.
  • A square bracket "[" or "]" shows that the endpoint is included.
  • The symbol \(-\infty\) always comes with a round bracket because infinity is not a number we can "reach" or include.
Understanding interval notation makes it easier to handle inequalities and represent solutions graphically.
Compound Inequalities
Compound inequalities involve two or more simple inequalities that are combined into a single expression. You can think of them as extending the concept of regular inequalities.
There are two main types:
  • "And" Compound Inequalities: These require both conditions to be satisfied simultaneously. For example, \(x > -3\) and \(x \leq 5\) would intersect or "AND" to give a range \((-3, 5]\).
  • "Or" Compound Inequalities: These work if at least one of the conditions is satisfied. For example, \(x < -1\) or \(x \geq 2\).
In our example, where the inequality simplifies to \(y \leq \frac{1}{8}\), it's straightforward because it鈥檚 a single inequality. However, when dealing with compound inequalities, you need to consider how the intersection or union of multiple sets affects the solution.
Graphing Inequalities
Graphing inequalities visually represents the solution set on a number line, adding clarity to the algebraic expression.
Steps to effectively graph an inequality like \(y \leq \frac{1}{8}\):
  • Draw a horizontal line to represent the number line.
  • Identify and mark the critical number, in this case, \(\frac{1}{8}\).
  • Since the inequality includes \(\frac{1}{8}\), place a "closed" dot on \(\frac{1}{8}\) to show it is part of the solution set.
  • Shade the region on the number line to the left of \(\frac{1}{8}\) to indicate that all numbers less than \(\frac{1}{8}\) are included.
Graphing helps in quickly identifying the range of solutions and verifying your solution visually.

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