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Chapter 2: Problem 10

Solve the inequality and specify the answer using interval notation. $$\frac{2}{3} \leq \frac{5-3 t}{-2} \leq \frac{3}{4}$$

Short Answer

Expert verified
The solution is \((-\infty, \frac{13}{6}]\).

Step by step solution

01

Understand the Structure

The given inequality is a compound inequality with two parts: \( \frac{2}{3} \leq \frac{5-3t}{-2} \) and \( \frac{5-3t}{-2} \leq \frac{3}{4} \). We need to solve each part separately and find the intersection of the solutions.
02

Isolate the Variable for the First Inequality

Start with \( \frac{2}{3} \leq \frac{5-3t}{-2} \). Multiply all terms by -2 (remember to reverse the inequality signs when multiplying by a negative number):\[-2 \cdot \frac{2}{3} \geq 5 - 3t\], giving \(-\frac{4}{3} \geq 5 - 3t\).
03

Step 2.1: Simplify the First Inequality

Rearrange to isolate \(t\) by subtracting 5 from both sides:\[-\frac{4}{3} - 5 \geq -3t\], which simplifies to \(-\frac{19}{3} \geq -3t\).
04

Step 2.2: Solve for t in the First Inequality

Divide through by -3 and reverse the inequality sign:\[t \leq \frac{19}{9}\].
05

Isolate the Variable for the Second Inequality

Now solve \( \frac{5-3t}{-2} \leq \frac{3}{4} \). Again, multiply all terms by -2 (reversing the inequality):\[-2 \cdot \frac{5-3t}{-2} \geq 3/4 \cdot -2\], giving \(5 - 3t \geq -\frac{3}{2}\).
06

Step 3.1: Simplify the Second Inequality

Rearrange to isolate \(t\) by subtracting 5:\(-3t \geq -\frac{3}{2} - 5\), which simplifies to \(-3t \geq -\frac{13}{2}\).
07

Step 3.2: Solve for t in the Second Inequality

Divide through by -3, remembering to reverse the inequality:\[t \leq \frac{13}{6}\].
08

Combine the Solutions

The solutions to the combined inequalities are \(t \leq \frac{19}{9}\) and \(t \leq \frac{13}{6}\). Since \(\frac{19}{9} > \frac{13}{6}\), the more restrictive condition is \(t \leq \frac{13}{6}\).
09

Write the Solution in Interval Notation

Since \(t\) must be less than or equal to \(\frac{13}{6}\), the interval notation is:\((-\infty, \frac{13}{6}]\).

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Key Concepts

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

Compound Inequalities
Compound inequalities are mathematical statements that involve two separate inequalities joined by the word "and" or "or." The inequality presented in the exercise is of the type "and," meaning both conditions must be true simultaneously. In our exercise, we deal with two parts: \( \frac{2}{3} \leq \frac{5-3t}{-2} \) and \( \frac{5-3t}{-2} \leq \frac{3}{4} \). Solving a compound inequality involves tackling each inequality separately and then finding the intersection of their solution sets.When working with compound inequalities:
  • Understand that you are making a statement about a single variable that must satisfy both conditions.
  • The solution to the inequality is the overlapping set of solutions between the two conditions.
  • For "and" inequalities, the intersection, or the overlap, of the two solution sets is what you'll be looking for.
In this problem, once we solve each part separately, we combine them to determine the range of values for \( t \) that satisfy both inequalities. This process ensures that the solution is a complete one that meets all given conditions.
Interval Notation
Interval notation provides a way to describe sets of numbers on the number line. It is particularly useful in expressing the solution to inequalities. There are two types of brackets used in interval notation:
  • Round brackets \( () \) indicate that an endpoint is not included in the interval.
  • Square brackets \( [] \) signify that an endpoint is included in the interval.
For example, the interval \((a, b)\) represents all numbers between \(a\) and \(b\), not including \(a\) and \(b\) themselves. On the other hand, \([a, b]\) would include both \(a\) and \(b\).
In this exercise, we determine that the solution does not extend beyond \( t \leq \frac{13}{6} \). Therefore, in interval notation, this is expressed as \((-\infty, \frac{13}{6}]\), indicating that all numbers from negative infinity up to and including \( \frac{13}{6} \) form our solution set.
Solving Inequalities
Solving inequalities requires understanding their properties and how they differ from solving equations. When solving inequalities, follow these steps:1. **Simplify each part of the inequality:** Treat each inequality separately to isolate the variable on one side.2. **Follow the rules for algebraic manipulation:** When you add, subtract, multiply, or divide both sides by a positive number, the inequality remains unchanged. However, when multiplying or dividing by a negative number, remember to flip the inequality sign. For instance, \(-3t \geq 3\) becomes \(t \leq -1\) when dividing by \(-3\).3. **Combine solutions:** After solving each part, bring the solutions together to determine the combined solution set for the compound inequality.Throughout this exercise, solving each portion carefully, flipping inequality signs when necessary, and being precise with the arithmetic ensures the correct solution. By doing so, our final answer accurately reflects the range of values that satisfy the original inequality conditions.

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Most popular questions from this chapter

Solve the inequalities. Suggestion: A calculator may be useful for approximating key numbers. $$\frac{x^{2}+3 x}{x^{2}+8 x+15}<0$$

Use the discriminant to determine how many real roots each equation has. $$x^{2}+\sqrt{3} x+\frac{3}{4}=0$$

Let \(a, b,\) and \(c\) be nonnegative numbers. Follow steps (a) through (e) to show that $$\sqrt[3]{a b c} \leq \frac{a+b+c}{3}$$ with equality holding if and only if \(a=b=c\). This result is known as the arithmetic-geometric mean inequality for three numbers. (Applications are developed in the projects at the ends of Sections 4.6 and \(4.7 .\) ) (a) By multiplying out the right-hand side, show that the following equation holds for all real numbers \(A, B\) and \(C\) $$\begin{aligned} 3 A B C=& A^{3}+B^{3}+C^{3}-\frac{1}{2}(A+B+C) \\ & \times\left[(A-B)^{2}+(B-C)^{2}+(C-A)^{2}\right] \quad \quad (1) \end{aligned}$$ (b) Now assume for the remainder of this exercise that \(A\) \(B,\) and \(C\) are nonnegative numbers. Use equation (1) to explain why $$3 A B C \leq A^{3}+B^{3}+C^{3}\quad \quad (2)$$ (c) Make the following substitutions in inequality ( 2 ): \(A^{3}=a, B^{3}=b,\) and \(C^{3}=c .\) Show that the result can be written $$\sqrt[3]{a b c} \leq \frac{a+b+c}{3} \quad \quad (3)$$ (d) Assuming that \(a=b=c,\) show that inequality (3) becomes an equality. (e) Assuming \(\sqrt[3]{a b c}=\frac{a+b+c}{3},\) show that \(a=b=c\) Hint: In terms of \(A, B,\) and \(C,\) the assumption becomes \(A B C=\frac{A^{3}+B^{3}+C^{3}}{3} .\) Use this to substitute for \(A B C\) on the left-hand side of equation (1). Then use the resulting equation to deduce that \(A=B=C,\) and consequently \(a=b=c\)

Determine all of the real-number solutions for each equation. (Remember to check for extraneous solutions.) $$\sqrt{2 t+5}-\sqrt{8 t+25}+\sqrt{2 t+8}=0$$

In the United States over the years \(1980-2000\), sulfur dioxide emissions due to the burning of fossil fuels can be approximated by the equation $$y=-0.4743 t+24.086$$ where \(y\) represents the sulfur dioxide emissions (in millions of tons) for the year \(t\), with \(t=0\) corresponding to \(1980 .\) Source: This equation (and the equation in Exercise 48) were computed using data from the book Vital Signs 1999 Lester Brown et al. (New York: W. W. Norton \& \(\mathrm{Co} ., 1999\) ). (a) Use a graphing utility to graph the equation \(y=-0.4743 t+24.086\) in the viewing rectangle [0,25,5] by \([0,30,5] .\) According to the graph, sulfur dioxide emissions are decreasing. What piece of information in the equation \(y=-0.4743 t+24.086\) tells you this even before looking at the graph? (b) Assuming this equation remains valid, estimate the year in which sulfur dioxide emissions in the United States might fall below 10 million tons per year. (You need to solve the inequality \(-0.4743 t+24.086 \leq 10 .)\)
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