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Chapter 4: Problem 104

(a) Mark on the coordinate line all those points \(x\) for which two of the following inequalities are true, and five are false: $$ |x|>1,|x|>2,|x|>3,|x|>4,|x|>5,|x|>6,|x|>7 $$ (b) Mark on the coordinate line all those points \(x\) for which two of the following inequalities are true, and five are false: $$ |x-1|>1,|x-2|>2,|x-3|>3,|x-4|>4,|x-5|>5,|x-6|>6,|x-7|>7 $$

Short Answer

Expert verified
(a) Mark x < -2 or x > 2; (b) Mark x < 0 or x > 4.

Step by step solution

01

Analyze the Inequalities (Part a)

The inequalities given are of the form \(|x|>n\). This means that the distance of \(x\) from the origin must be greater than \(n\). This implies that \(x>n \text{ or } x<-n\).
02

Select Two True Inequalities (Part a)

We need to find two inequalities that can be satisfied at the same time. To solve this, recognize that the smallest values of \(n\) will give us the largest range of truth. Let's pick the inequalities \(|x|>1\) and \(|x|>2\).
03

Solve the Two Inequalities (Part a)

For \(|x|>1\), the solution is \(x>1 \text{ or } x<-1\). For \(|x|>2\), the solution is \(x>2 \text{ or } x<-2\). Combining these, the overall solution is \(x>2 \text{ or } x<-2\). This means \(x\) must be less than \(-2\) or greater than \(2\).
04

Analyze the Inequalities (Part b)

The inequalities are of the form \(|x-n|>m\). This implies that \(x\) is at a distance greater than \(m\) from \(n\), which means the interval \(x>n+m \text{ or } x<n-m\).
05

Select Two True Inequalities (Part b)

Similar to Part a, we choose inequalities with the smallest values for true states. Let's pick \(|x-1|>1\) and \(|x-2|>2\).
06

Solve the Two Inequalities (Part b)

For \(|x-1|>1\), the solution is \(x>2\) or \(x<0\). For \(|x-2|>2\), the solution is \(x>4\) or \(x<0\). The intersection of these is \(x<0\) or \(x>4\). Thus, \(x\) must be less than \(0\) or greater than \(4\).
07

Mark Points on Coordinate Line

For Part a, mark points beyond \(2\) or before \(-2\). For Part b, mark points beyond \(4\) or before \(0\) on the coordinate line.

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Key Concepts

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

Absolute Value
Absolute value is a fundamental concept in mathematics that is used to describe the distance of a number from zero on the number line. It is denoted as \(|x|\), where \(x\) is any real number. The absolute value of \(x\), therefore, is always a non-negative number. This means that whether \(x\) is positive or negative, its absolute value will be without a sign.

For example, if \(x = 3\), then \(|x| = 3\); similarly, if \(x = -3\), then \(|x| = 3\) as well. This property makes absolute value particularly useful when comparing the magnitude of numbers without regard to their sign. It essentially measures how far a number is from zero on the number line, emphasizing distance and ignoring the direction.

Absolute values often appear in various mathematical expressions and equations, especially inequalities that describe ranges or limits within which certain conditions hold true. Understanding absolute value as a measure of distance helps solve problems related to range and limits by focusing on the magnitude rather than the direction of numbers.
Coordinate Line
A coordinate line, commonly known as a number line, is a visual representation of numbers in a linear format. This line aids in understanding numerical relationships and spatial orientation of numeric values.

On the coordinate line, numbers are placed at equal intervals, with a central point representing zero. Positive numbers are positioned to the right of zero, while negative numbers are represented to the left. Each point on this line corresponds directly to a real number.

This visual tool is especially beneficial when solving equations or inequalities involving absolute values. By marking specific points, you can visually observe intervals where certain conditions (such as inequalities) are true or false. For example, by marking points where absolute values satisfy specific conditions, one can easily represent all solutions and gain intuitive insight into the problem.
Distance from Origin
Distance from the origin refers to how far away a number is from zero on the number line. This concept is crucial when dealing with absolute values since it provides a clear, visual understanding of inequalities like \(|x| > n\).

The origin is essentially the zero point, or the middle point of the number line. When we refer to the distance from the origin, we signify the absolute value of the number. For a number \(x\), the expression \(|x| > n\) indicates that \(x\) is more than \(n\) units away from the origin. This can mean either being greater than \(n\) or less than \(-n\).

The concept is not just limited to positive or negative distances but emphasizes a range of possible values that satisfy the inequality. This is why understanding distance from the origin can simplify interpreting and solving these types of mathematical problems.
Number Line
A number line is an essential tool in learning and visualizing basic arithmetic operations, the positions of numbers, and inequalities. It represents real numbers in a horizontal line, with zero in the center, positive numbers extending to the right, and negative numbers stretching to the left.

Number lines are incredibly useful for solving inequalities because they provide a visual representation of how numbers interact in arithmetic. They can illustrate the span where solutions to certain inequalities lie. For instance, if we have an inequality \(|x| > 3\), marking regions on the number line can show where \(x\) would satisfy this condition, i.e., beyond \(3\) and \(-3\).

The visual aspect of number lines helps clarify the relationships and the magnitude of differences between numbers, supporting a deeper understanding of how inequalities reflect conditions in real-world contexts. They are a simple yet powerful means to represent and solve problems that involve ranges or limits, such as those seen in absolute value problems.

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Most popular questions from this chapter

The two hundred numbers $$ 1,2,3,4,5, \ldots, 200 $$ are written on the board. Students take turns to replace two numbers \(a, b\) from the current list by their sum divided by \(\sqrt{2}\). Eventually one number is left on the board. Prove that the final number must be less than \(2000 . \quad \triangle\)

(a) Given the equation \(x^{3}+3 x^{2}-4=0,\) choose a constant \(a,\) and then change variable by substituting \(y=x+a\) to produce an equation of the form \(y^{3}+k y=\) constant. (b) In general, given any cubic equation \(a x^{3}+b x^{2}+c^{x}+d=0\) with \(a \neq 0\), show how to change variable so as to reduce this to a cubic equation with no quadratic term.

Solve the following systems of simultaneous equations. (a)(i) \(x+y=1, \quad y+z=2, \quad x+z=3\) (ii) \(u v=2, \quad v w=4, \quad u w=8\) (b) (i) \(x+y=2, \quad y+z=3, \quad x+z=4\) (ii) \(u v=6, \quad v w=10, \quad u w=15\) (iii) \(u v=6, \quad v w=10, \quad u w=30\) (iv) \(u v=4, \quad v w=8, \quad u w=16\)

(a) Derive a closed formula for the sum of the geometric series $$ 1+r+r^{2}+r^{3}+\cdots+r^{n} $$ (The meaning of closed formula was discussed in the Note to the solution to Problem \(54(b)\) in Chapter \(2 .)\) (b) Derive a closed formula for the sum of the geometric series $$ a+a r+a r^{2}+a r^{3}+\cdots+a r^{n} . $$ We started this subsection by looking for prime numbers of the form \(x^{2}-1\). A simple-minded approach to the distribution of prime numbers might look for formulae that generate primes \(-\) all the time, or infinitely often, or at least much of the time. In Chapter 1 (Problem 25) you showed that no prime of the form \(4 k+3\) can be "represented" as a sum of two squares (i.e. in the form " \(\left.x^{2}+y^{2 \prime \prime}\right)\), and we remarked that every other prime can be so represented in exactly one way. It is true (but not obvious) that roughly half the primes fall into the second category; so it follows that substituting integers for the two variables in the polynomial \(x^{2}+y^{2}\) produces a prime number infinitely often.

The equation \(x^{3}+3 x^{2}-4=0\) clearly has " \(x=1\) " as a positive solution. (The other two solutions are \(x=-2,\) and \(x=-2-\) a repeated root; however negatives were viewed with suspicion in the sixteenth century, so this root might well have been ignored.) Try to understand how the following sequence of moves "finds the root \(x=1 "\) : (i) substitute \(y=x+1\) to get a cubic equation in \(y\) with no term in \(y^{2}\); (ii) imagine \(y=u+v\) and interpret the identity for $$ (u+v)^{3}=u^{3}+3 u v(u+v)+v^{3} $$ as your cubic equation in \(y\); (iii) solve the simultaneous equations " \(3 u v=3 ", " u^{3}+v^{3}=2\) " (not by guessing, but by substituting \(v=\frac{1}{u}\) from the first equation into the second to get a quadratic equation in " \(u^{3}\) ", which you can then solve for \(u^{3}\) before taking cube roots); (iv) then find the corresponding value of \(v\), hence the value of \(y=u+v,\) and hence the value of \(x\). The simple method underlying Problem 135 is in fact completely general. Given any cubic equation $$ a x^{3}+b x^{2}+c x+d=0 \quad \text { (with } \left.a \neq 0\right) $$ we can divide through by \(a\) to reduce this to $$ x^{3}+p x^{2}+q x+r=0 $$ with leading coefficient \(=1 .\) Then we can substitute \(y=x+\frac{p}{3}\) and reduce this to a cubic equation in \(y\) $$ y^{3}-3\left(\frac{p}{3}\right)^{2} y+q y+\left[r+2\left(\frac{p}{3}\right)^{3}-q\left(\frac{p}{3}\right)\right]=0 $$ which we can treat as having the form $$ y^{3}-m y-n=0 $$ So we can set \(y=u+v\) (for some unknown \(u\) and \(v\) yet to be chosen), and treat the last equation as an instance of the identity $$ (u+v)^{3}-3 u v(u+v)-\left(u^{3}+v^{3}\right)=0 $$ which it will become if we simply choose \(u\) and \(v\) to solve the simultaneous equations $$ 3 u v=m, \quad u^{3}+v^{3}=n . $$ We can then solve these equations to find \(u,\) then \(v-\) and hence find \(y=u+v\) and \(x=y-\frac{p}{3}\).
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