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Chapter 8: Problem 55

Graph the solution set, and write it using interval notation. $$ 4 \leq-9 x+5<8 $$

Short Answer

Expert verified
The solution set in interval notation is \((-\frac{1}{3}, \frac{1}{9}]\).

Step by step solution

01

- Break the Compound Inequality

Separate the compound inequality into two individual inequalities: 1) \(4 \leq -9x+5\)2) \(-9x+5 < 8\)
02

- Solve the First Inequality

Or equivalently: \(x \leq \frac{1}{9}\)
03

- Solve the Second Inequality

Solve the second inequality: \(-9x + 5 < 8\)Subtract 5 from both sides: \(-9x + 5 - 5 < 8 - 5\)Simplify: \(-9x < 3\)Divide by -9 (remember to reverse the inequality sign when dividing by a negative number): \(x > \frac{3}{-9}\)This simplifies to: \(x > -\frac{1}{3}\)
04

- Combine the Solutions

Combine the solutions: \(-\frac{1}{3} < x \leq \frac{1}{9}\)
05

- Write the Solution in Interval Notation

Express the solution set in interval notation: \((-\frac{1}{3}, \frac{1}{9}]\)
06

- Graph the Solution

Graph the interval on the number line: - Open circle at \(-\frac{1}{3}\) to indicate \(x > -\frac{1}{3}\), and filled circle at \(\frac{1}{9}\) to indicate \(x \leq \frac{1}{9}\). Shade the region between them.

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Key Concepts

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

solving inequalities
Inequalities express a relationship where one side is not necessarily equal to the other side. In algebra, solving inequalities involves finding the values of a variable that make the inequality true. Let's break down the steps to solve the compound inequality given: \( 4 \leq -9x + 5 < 8 \).
  • First, you split the compound inequality into two separate inequalities: \(4 \leq -9x + 5 \) and \(-9x + 5 < 8 \).

  • Then, solve each part individually. For \( 4 \leq -9x + 5 \):
    1. Subtract 5 from both sides to get: \( -1 \leq -9x \).
    2. Divide by -9 (note: reversing the inequality sign): \( x \leq \frac{1} {9} \).

  • Similarly, for \( -9x + 5 < 8 \):
    1. Subtract 5 from both sides: \( -9x < 3 \).
    2. Divide by -9 (note: reversing the inequality sign): \( x > -\frac{1}{3} \).

  • Finally, combine the two solutions to get \( -\frac{1}{3} < x \leq \frac{1}{9} \).

interval notation
Interval notation is a way of writing sets of numbers that fall within a certain range. It uses brackets and parentheses to describe the endpoints of an interval. For example, \( [a, b] \) includes endpoints \(a\) and \(b\), whereas \( (a, b) \) excludes them.
In this exercise, the solution \( -\frac{1}{3} < x \leq \frac{1}{9} \) needs to be written in interval notation.
  • We use a parenthesis \((\) to indicate that \( -\frac{1}{3}\) is not included in the solution.
  • We use a bracket \(] \) to show that \( \frac{1}{9} \) is included in the solution.
  • Therefore, the interval notation for the solution is \( ( -\frac{1}{3}, \frac{1}{9}] \).

  • This interval notation tells us exactly where the solutions for the inequality lie on a number line.
graphing solutions
Graphing the solution of an inequality involves drawing it on a number line to visually represent the range of possible solutions. Here鈥檚 how to graph the given solution:\( ( -\frac{1}{3}, \frac{1}{9} ] \)
  • Plot a number line marking key points, specifically \( -\frac{1}{3} \) and \( \frac{1}{9} \).
  • Use an open circle on \( -\frac{1}{3} \) to indicate that \( -\frac{1}{3} \) is not part of the solution (because we use \( < \), not \( \leq\)).
  • Use a filled circle on \( \frac{1}{9} \) to show that \( \frac{1}{9} \) is included in the solution (due to \( \leq \)).
  • Shade the region between \( -\frac{1}{3} \) and \( \frac{1}{9} \) to represent all values that \( x \) can take.

This visual graph makes it easier to understand the solution set and clearly shows the range of numbers that satisfy the inequality.

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Most popular questions from this chapter

Graph the solution set, and write it using interval notation. $$ -(4+r)+2-3 r<-14 $$

In later courses in mathematics, it is sometimes necessary to find an interval in which \(x\) must lie in order to keep y within a given difference of some number. For example, suppose $$y=2 x+1$$ and we want \(y\) to be within 0.01 unit of \(4 .\) This criterion can be written as $$|y-4|<0.01$$ Solving this inequality shows that \(x\) must lie in the interval (1.495,1.505) to satisfy the requirement. Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=4 x-8,\) and the difference of \(y\) and 3 is less than 0.001

Graph the solution set, and write it using interval notation. $$ 2 x-8 \geq-2 x $$

Graph the solution set, and write it using interval notation. $$ x-3(x+1) \leq 4 x $$

Graph the solution set, and write it using interval notation. $$ -3 \leq \frac{3 x+1}{4} \leq 3 $$
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