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Chapter 2: Problem 4

Solve the inequality and specify the answer using interval notation. $$1-2(t+3)-t \leq 1-2 t$$

Short Answer

Expert verified
The solution in interval notation is \([-6, \infty)\).

Step by step solution

01

Simplify the Expression

First, expand the expression on the left side of the inequality. The given inequality is \(1 - 2(t+3) - t \leq 1 - 2t\). Start by distributing the \(-2\) in \(-2(t+3)\):\[1 - 2t - 6 - t \leq 1 - 2t\].Combine like terms on the left side: \(1 - 2t - 6 - t = -2t - t + 1 - 6 = -3t - 5\). The inequality now reads:\[-3t - 5 \leq 1 - 2t\].
02

Isolate the Variable Terms

Our goal is to isolate \(t\) on one side. Start by adding \(3t\) to both sides of the inequality:\[-3t - 5 + 3t \leq 1 - 2t + 3t\].Simplifying both sides gives:\[-5 \leq 1 + t\].
03

Solve for the Variable

Now isolate \(t\) by subtracting 1 from both sides:\[-5 - 1 \leq t\].Which simplifies to:\[-6 \leq t\].
04

Write in Interval Notation

The inequality \(-6 \leq t\) means that \(t\) is greater than or equal to \(-6\). In interval notation, this is written as:\([-6, \infty)\).

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Key Concepts

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

Solving Inequalities
To solve inequalities, we follow a process similar to solving equations, but with a few important differences. When you solve an inequality like \(1 - 2(t+3) - t \leq 1 - 2t\), you aim to isolate the variable (in this case, \(t\)) on one side. This involves several steps, which typically include:
  • Expanding: Break down any brackets by distributing the numbers outside the parentheses across the terms inside. This helps to combine like terms.

  • Combining like terms: This step involves simplifying expressions on each side of the inequality by adding and subtracting similar terms. It simplifies comparisons.

  • Isolating the variable: Use basic algebraic techniques like adding, subtracting, multiplying, or dividing to get the variable by itself.

    • \
Importantly, remember that if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This is a key concept that distinguishes inequalities from equations.
Interval Notation
Interval notation is a concise way to describe sets of numbers that satisfy an inequality. This notation uses parentheses \(()\) and square brackets \([]\) to indicate different types of containment:
  • Square Brackets [ ]: These are used when the endpoint is included in the set, known as a closed interval. For example, \([-6, \, \infty)\) includes the number \(-6\) itself.

  • Parentheses ( ): These indicate that the endpoint is not included, usually used in open intervals. In \([-6, \, \infty)\), \(\infty\) does not represent a specific number, so we use a parenthesis.

Interval notation thus provides a clearer, more compact way to express inequalities. In the example \(-6 \leq t\), it means that \(t\) can be any number from \(-6\) to infinity, including \(-6\), hence written as \([-6, \, \infty)\).
Algebraic Manipulation
Algebraic manipulation involves using operations to transform expressions and solve equations or inequalities. For solving the inequality \(1 - 2(t+3) - t \leq 1 - 2t\), you perform several algebraic steps such as:
  • Distributing factors: Apply distribution laws, like distributing \(-2\) across \(t+3\) to simplify terms.

  • Combining terms: Group and simplify terms to reduce the complexity of expressions. For example, combine \(-2t\) and \(-t\) into \(-3t\).

  • Maintaining equality/inequality: Carefully add, subtract, multiply, or divide terms while maintaining the inequality, systematically moving variables and constants to different sides as needed.

Such manipulations help in systematically reducing the expression to a simple inequality such as \(-6 \leq t\), aiding in understanding and solving the problem effectively. These techniques form the backbone of algebra and are essential for tackling more complex mathematical problems.

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Most popular questions from this chapter

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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