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Chapter 2: Problem 12

Solve the inequality and specify the answer using interval notation. $$\frac{9}{10} < \frac{3 x-1}{-2} < \frac{91}{100}$$

Short Answer

Expert verified
The solution is \((-\frac{41}{150}, -\frac{4}{15})\) in interval notation.

Step by step solution

01

Simplify the Inequality

The inequality \(\frac{9}{10} < \frac{3x-1}{-2} < \frac{91}{100}\) consists of two parts to solve: 1. \(\frac{3x-1}{-2} > \frac{9}{10}\) 2. \(\frac{3x-1}{-2} < \frac{91}{100}\). We will begin by solving the first part of the inequality.
02

Solve Part 1 of the Inequality

For \(\frac{3x-1}{-2} > \frac{9}{10}\), multiply both sides by \(-2\) and reverse the inequality sign to get:\[3x - 1 < -\frac{18}{10}\]Simplify to obtain:\[3x - 1 < -\frac{9}{5}\]Add 1 to both sides:\[3x < -\frac{9}{5} + 1\]Convert \(1\) to \(\frac{5}{5}\) to combine:\[3x < -\frac{4}{5}\]Divide by \(3\) to solve for \(x\):\[x < -\frac{4}{15}\].
03

Solve Part 2 of the Inequality

Now solve \(\frac{3x-1}{-2} < \frac{91}{100}\). Multiply both sides by \(-2\) and reverse the inequality to obtain:\[3x - 1 > -\frac{182}{100}\]Simplify the right side:\[3x - 1 > -\frac{91}{50}\]Add 1 to both sides:\[3x > -\frac{91}{50} + 1\]Convert \(1\) to \(\frac{50}{50}\) for addition:\[3x > -\frac{41}{50}\]Divide by \(3\):\[x > -\frac{41}{150}\].
04

Combine Solutions

Combine the solutions from Steps 2 and 3: \(x > -\frac{41}{150}\) and \(x < -\frac{4}{15}\).These combined inequalities are written as:\[-\frac{41}{150} < x < -\frac{4}{15}\].
05

Write Solution in Interval Notation

The solution in interval notation is:\[(-\frac{41}{150}, -\frac{4}{15})\].

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Key Concepts

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

Understanding Interval Notation
Interval notation is a mathematical way to describe a range of numbers along a number line. It's useful for expressing solutions to inequalities in a concise manner. In interval notation:
  • Parentheses, like \(\left(\right)\), indicate that an end of the interval is not included, meaning the set of solutions does not touch this boundary.
  • Brackets, like \[\left[\right]\], indicate that an endpoint is included in the set of solutions.
For instance, \( (a, b) \) means all numbers between \(a\) and \(b\), but not \(a\) and \(b\) themselves. In our example, the solution is represented as \( (-\frac{41}{150}, -\frac{4}{15}) \), meaning all values \(x\) between \(-\frac{41}{150}\) and \(-\frac{4}{15}\), excluding the endpoints. Each inequality solution provides a specific interval notation, helping students and mathematicians communicate solutions precisely.
Explaining Compound Inequalities
Compound inequalities involve two separate inequalities joined together by "and" or "or." They describe multiple conditions that a solution must satisfy and are usually written in the form: \( a < x < b \).
  • An "and" compound inequality describes solutions that satisfy both conditions simultaneously. When solving, you find values of \(x\) that make both inequalities true at the same time. In interval notation, this is represented simply as a segment on the number line.
  • An "or" compound inequality provides solutions if either condition is met. This can be thought of as a union of two separate intervals.
In our exercise, the compound inequality \( -\frac{41}{150} < x < -\frac{4}{15} \) indicates that \(x\) is greater than \(-\frac{41}{150}\) and less than \(-\frac{4}{15}\), meaning \(x\) must satisfy both conditions together.
Mastering Inequality Solution Steps
Solving inequalities involves a series of systematic steps similar to solving equations, but with some additional rules to consider. These steps are necessary to isolate the variable and determine the solution range.
  • Simplify the inequality: Break down the complex inequality into parts if it's compound, then deal with each part individually.
  • Reverse inequality signs: If you multiply or divide by a negative number, don't forget to reverse the inequality sign. This is crucial to preserving the inequality's truth.
  • Combine solutions: After solving each part of the inequality, combine the solutions logically to form a compound inequality, if necessary.
  • Express in interval notation: Reflect the solution on a number line and use interval notation to precisely communicate the set of solutions.
Through these steps, you can transition from solving the parts of an inequality to expressing a clear, logical answer. Our example showed how multiplying by \-2\ reversed the inequality signs, leading us to the final solution in interval notation: \( (-\frac{41}{150}, -\frac{4}{15}) \). Understanding every step ensures accuracy and clarity in finding the solution.

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Most popular questions from this chapter

Suppose that after studying a corporation's records, a business analyst predicts that the corporation's monthly revenues \(R\) for the near future can be closely approximated by the equation $$\begin{aligned}R=&-0.0217 x^{5}+0.626 x^{4}-6.071 x^{3}+25.216 x^{2} \\\&-57.703 x+159.955 \quad(1 \leq x \leq 12)\end{aligned}$$ where \(R\) is the revenue (in thousands of dollars) for the month \(x,\) with \(x=1\) denoting January, \(x=2\) denoting February, and so on. (a) According to this model, for which months will the monthly revenue be no more than \(\$ 80,000 ?\) Hint: You need to solve the inequality \(R \leq 80 .\) Round each key number to the nearest integer. (Why?) (b) For which months, if any, will the monthly revenue be at least \(\$ 120,000 ?\)

Show that the quadratic equation $$(x-p)(x-q)=r^{2} \quad(p \neq q)$$ has two distinct real roots.

(a) On the same set of axes, graph the equations \(y=x^{2}+8 x+16\) and \(y=x^{2}-8 x+16\) (b) Use the graphs to estimate the roots of the two equations \(x^{2}+8 x+16=0\) and \(x^{2}-8 x+16=0 .\) How do the roots appear to be related? (c) Solve the two equations in part (b) to determine the exact values of the roots. Do your results support the response you gave to the question at the end of part (b)?

Solve the inequalities Suggestion: A calculator may be useful for approximating key numbers. $$4\left(x^{2}-9\right)-\left(x^{2}-9\right)^{2}>-5$$

Solve the inequalities. Suggestion: A calculator may be useful for approximating key numbers. $$\frac{x^{2}-1}{x^{2}+8 x+15} \geq 0$$
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