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Chapter 2: Problem 18

Solve the inequality and specify the answer using interval notation. (a) \(|t| \geq 0\) (b) \(|t| \leq 0\)

Short Answer

Expert verified
(a) \((-\infty, +\infty)\); (b) \([0, 0]\)

Step by step solution

01

Understanding Absolute Value

The absolute value of a number refers to its distance from zero on the number line without considering the direction. Therefore, for any real number, the absolute value is non-negative.
02

Analyzing Absolute Value Inequality (a)

Given the inequality \(|t| \geq 0\), we must determine when the absolute value of a number is greater than or equal to zero. \(|t|\) is always non-negative, and hence, it always satisfies \(|t| \geq 0\) for all real numbers \(t\).
03

Solution for Part (a)

Since all real numbers satisfy the condition \(|t| \geq 0\), the solution is all real numbers. In interval notation, this is expressed as \((-\infty, +\infty)\).
04

Analyzing Absolute Value Inequality (b)

Given the inequality \(|t| \leq 0\), we must determine when the absolute value of a number can be less than or equal to zero. Since \(|t|\) is non-negative, the only time it equals zero is when \(t = 0\). Hence, the solution is when \(t = 0\).
05

Solution for Part (b)

Since \(|t| = 0\) only when \(t = 0\), the solution in interval notation is simply \([0, 0]\), which represents the single number zero.

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Key Concepts

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

Absolute Value
The absolute value of a number is a very useful concept in mathematics. It essentially conveys how far a number is from zero on the number line 鈥� without caring whether it's to the left or right. For example, the absolute value of both \(5\) and \(-5\) is \(5\). This is expressed mathematically as \(|t|\), which means "absolute value of \(t\)".
The absolute value is always non-negative:
  • For positive numbers, \(|t| = t\)
  • For negative numbers, \(|t| = -t\)
  • For zero itself, \(|0| = 0\)
Understanding this concept helps solve inequalities involving absolute values, like determining conditions where \(|t|\) is greater or less than a certain number.
Interval Notation
Interval notation is a concise way of expressing the set of solutions for inequalities. It's like drawing a map where numbers on a number line are highlighted without drawing the full line.
It's composed of two numbers which represent the endpoints of the solutions:
  • The left endpoint signifies the smallest number in the interval.
  • The right endpoint is the greatest number in the interval.
Consider two common types of brackets:
  • Square brackets \[ [ ] \] indicate inclusive boundaries. For example, \[ [0, 0] \] means only \({0}\) is included in the set.
  • Parentheses \( ( ) \) are used for open endpoints, meaning the numbers are not part of the solution. For instance, \( (-\infty, +\infty) \) implies all real numbers, as infinity can't be reached or included.
Real Numbers
Real numbers form the comprehensive set of all numbers we typically work with. They include:
  • Natural numbers like \(1, 2, 3,...\)
  • Whole numbers like \(0, 1, 2,...\)
  • Integers, which incorporate both positive and negative numbers such as \(-3, -2, -1, 0, 1, 2, 3,...\)
  • Rational numbers, numbers that can be represented by a fraction \(\frac{a}{b}\) with integers \(a, b\)and \(b eq 0\)
  • Irrational numbers, numbers that can鈥檛 be written as a simple fraction (e.g., \(\pi\) and \(\sqrt{2}\))
These real numbers can be thought of as all the points along the infinitely stretching number line without any breaks.
Mathematical Notation
Mathematical notation is a universal language that distills complex mathematical ideas into symbols and expressions. This form is universally understood and allows clear communication:
  • Operators such as \(+, -, \times, \div\) describe arithmetic actions.
  • Inequality symbols \(<, \leq, >, \geq\) compare sizes and magnitudes.
  • Specific symbols like the absolute value lines \(| |\) convey special meanings.
  • Notation is essential for defining functions, equations, and even sets, conveying a multitude of information crisply.
For example, \(t \leq 0\) and \(t \geq 0\) effectively communicate a host of conditions at a glance. Understanding these symbols and their proper use helps in decoding and solving mathematical problems.

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Most popular questions from this chapter

Solve the inequalities. Suggestion: A calculator may be useful for approximating key numbers. $$\frac{1}{x} \leq \frac{1}{x+1}$$

Assume that \(a\) and \(b\) are the roots of the equation \(x^{2}+p x+q=0\) (a) Find the value of \(a^{2} b+a b^{2}\) in terms of \(p\) and \(q\) Hint: Factor the expression \(a^{2} b+a b^{2}\) (b) Find the value of \(a^{3} b+a b^{3}\) in terms of \(p\) and \(q\) Hint: Factor. Then use the fact that \(a^{2}+b^{2}=(a+b)^{2}-2 a b\)

(a) Use a graph to estimate the solution set for each inequality. Zoom in far enough so that you can estimate the relevant endpoints to the nearest thousandth. (b) Exercises \(61-70\) can be solved algebraically using the techniques presented in this section. Carry out the algebra to obtain exact expressions for the endpoints that you estimated in part (a). Then use a calculator to check that your results are consistent with the previous estimates. $$x^{4}-2 x^{2}-1>0$$

Show that the quadratic equation $$a x^{2}+b x-a=0 \quad(a \neq 0)$$ has two distinct real roots.

(a) Use the quadratic formula to show that the roots of the equation \(x^{2}+3 x+1=0\) are \(\frac{1}{2}(-3 \pm \sqrt{5})\) (b) Show that \(\frac{1}{2}(-3+\sqrt{5})=1 /\left[\frac{1}{2}(-3-\sqrt{5})\right]\) Hint: Rationalize the denominator on the right-hand side of the equation. (c) The result in part (b) shows that the roots of the equation \(x^{2}+3 x+1=0\) are reciprocals. Can you find another, much simpler way to establish this fact?
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