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Chapter 4: Problem 10

Show that \(|a-b| \leq c\) is equivalent to \(b-c \leq a \leq b+c\).

Short Answer

Expert verified
Both inequalities express the same range for possible values of \(a\), hence they are equivalent.

Step by step solution

01

Definition of Absolute Value

To start, recall the definition of absolute value. For any real numbers, the absolute value inequality \(|x| \leq c\) implies that \(-c \leq x \leq c\).

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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 its distance from zero on the number line, regardless of direction. It is always non-negative. Absolute value is denoted by vertical bars, for example, \(|x|\).
When you have \(|a-b|\), it represents the distance between the numbers \(a\) and \(b\).
The inequality \(|a-b| \leq c\) means that this distance is less than or equal to \(c\).
This concept helps us to understand situations without worrying about which number is larger, \(a\) or \(b\).
Real Numbers
Real numbers include both rational and irrational numbers. They represent a continuous and unbroken line of values. Any point on the number line is a real number, like \(\pi\), \(-3\), or \(0.75\).
In the context of this exercise, both \(a\) and \(b\) are considered real numbers. This means that any calculation or manipulation involving these variables proceeds with the assumption that they can be any value on the number line, not just integers or fractions.
Understanding this allows us to apply the properties of absolute value and inequality broadly.
Inequality
Inequalities express a relationship where one value is not equal to another, indicating that one is less than or greater than the other. The inequality \(b-c \leq a \leq b+c\) gives a range of values that \(a\) can take.
When working with this type of double inequality, you're describing all the possible values of \(a\) that satisfy both parts of the inequality.
In practice, understanding inequalities involves being able to manipulate and interpret these ranges, anticipating the set of numerical solutions that comply with given constraints.
Mathematical Proof
A mathematical proof provides a logical argument that a statement is true beyond any doubt. Proofs follow a series of steps based on axioms, definitions, and previously established statements.
The goal of this exercise is to show that \(|a-b| \leq c\) is equivalent to the double inequality \(b-c \leq a \leq b+c\). This process involves using the definition of absolute value and manipulating the expressions to demonstrate equality.
Such proofs assure us of the solution's correctness and deepen our understanding of how mathematical principles interconnect.

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Most popular questions from this chapter

Given the equation: $$ x^{2}+y^{2}=1 $$ (a) Transpose the equation to make \(y\) the subject of the transposed equation. (b) Construct ordered pairs of numbers corresponding to the integer values of \(x\) where \(-1 \leq x \leq 1\) in intervals of \(0.2\). (c) Plot the ordered pairs of numbers on a Cartesian graph and join the points plotted with a continuous curve.

Using a spreadsheet plot the graph of: $$ y=\frac{x^{3}}{1-x^{2}} $$ for \(-2 \leq x \leq 2 \cdot 6\) with a step value of \(0.2 .\) Draw a sketch of this graph on a sheet of graph paper indicating discontinuities and asymptotic behaviour more, accurately.

Given the equation: $$ \left(\frac{x}{2}\right)^{2}+\left(\frac{y}{4}\right)^{2}=1 $$ transpose it to find \(y\) in terms of \(x\). With the aid of a spreadsheet describe the shape that this equation describes.

Using a spreadsheet plot the graph of: \(y=x^{3}+10 x^{2}+10 x-1\) for \(-10 \leq x \leq 2\) with a step value of \(0 \cdot 5\)

Draw the graph of \(f(x)=|3 x-4|+2\) and describe the differences between that graph and the graph of \(g(x)=|4-3 x|+2\)
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