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Chapter 1: Problem 35

Solve the inequalities in Exercises \(35-42 .\) Express the solution sets as intervals or unions of intervals and show them on the real line. Use the result \(\sqrt{a^{2}}=|a|\) as appropriate. $$ x^{2}<2 $$

Short Answer

Expert verified
The solution is \((-\sqrt{2}, \sqrt{2})\).

Step by step solution

01

Understand the Inequality

The inequality given is \( x^2 < 2 \). This means we are looking for all values of \( x \) whose square is less than 2.
02

Solve Using Square Roots

Take the square root on both sides of the inequality \( x^2 < 2 \). We obtain \( |x| < \sqrt{2} \). The result tells us that \( x \) must be within the range of \( -\sqrt{2} < x < \sqrt{2} \).
03

Express the Solution as an Interval

The solution to the inequality can be expressed as the interval \((-\sqrt{2}, \sqrt{2})\). This represents all real numbers \( x \) that are greater than \(-\sqrt{2}\) and less than \(\sqrt{2}\).
04

Visualize on the Real Line

To show the solution on the real line, draw a line with open circles at \(-\sqrt{2}\) and \(\sqrt{2}\). Shade the region between the circles, indicating that all numbers within this range satisfy the inequality.

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Key Concepts

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

Square Roots
The square root of a number is a special value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 multiplied by itself equals 9. When solving inequalities like \( x^2 < 2 \), taking square roots helps determine the range of values that a variable can have.
In this context, it's important to understand:
  • Square roots provide a way to move from squared values back to their original counterparts.
  • The expression \( \sqrt{a^2} = |a| \) is crucial because it tells us that the square root of a squared value results in the absolute value of that original number.
  • This property allows us to express solutions involving inequalities as intervals on the number line.
Whenever you deal with squaring and square roots, remember their relationship as a fundamental part of understanding and solving inequalities efficiently.
Absolute Value
Absolute value is a mathematical concept that measures the distance of a number from zero on the real number line. It is always positive or zero, never negative. For example, the absolute value of both -3 and 3 is 3. This concept is particularly useful in understanding solutions that involve inequalities and square roots.
  • The absolute value of a number \( x \), written as \( |x| \), represents its magnitude without considering its direction (positive or negative).
  • It plays a significant role when working with square roots, as seen in the inequality \( x^2 < 2 \).
  • Through the transformation \( x^2 < 2 \) to \( |x| < \sqrt{2} \), we identify that \( x \) is restricted to being within a particular range on the real line.
Understanding absolute values helps in clearly defining these ranges and grasping interval-based solutions.
Real Line
The real line is a representation of all real numbers in a continuous, infinite line. Each point on this line corresponds to a unique real number. It is used extensively to visualize solutions to inequalities and intervals especially when showing which values satisfy a condition.
  • The numbers are arranged from left to right in increasing order, extending infinitely in both directions.
  • When expressing solutions like \( -\sqrt{2} < x < \sqrt{2} \) visually, we can shade the section of the real line between the points \(-\sqrt{2}\) and \(\sqrt{2}\).
  • Open or closed circles show whether endpoints are included or not. For instance, open circles at these points indicate that neither \(-\sqrt{2}\) nor \(\sqrt{2}\) are part of the solution.
Utilizing the real line for visual representation provides clarity and helps confirm our algebraic solutions.
Intervals
Intervals are a way of describing a set of numbers between two endpoints. They can be open, closed, or a combination, depending on whether the endpoints are included in the set. For the inequality \( x^2 < 2 \), the solution is expressed as an interval.
  • An open interval, \((a, b)\), includes all numbers between \(a\) and \(b\) but not \(a\) and \(b\) themselves. Thus \(( -\sqrt{2}, \sqrt{2} )\) means all numbers more than \(-\sqrt{2}\) and less than \(\sqrt{2}\).
  • Conversely, a closed interval \([a, b]\) includes the endpoints \(a\) and \(b\).
  • Intervals offer concise notation for describing solutions and constraints in mathematics, and are essential in expressing the range of valid solutions for inequalities.
By understanding intervals, we can easily communicate complex sets of solutions and visualize these solutions on the real line for better comprehension.

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Most popular questions from this chapter

Graph the functions in Exercises \(23-26\) $$ f(x)=\left\\{\begin{array}{ll}{x,} & {0 \leq x \leq 1} \\ {2-x,} & {1

Graph the upper branch of the hyperbola \(y^{2}-16 x^{2}=1\)

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=1 / x^{2} \quad \text { Left } 2, \text { down } 1 $$

In Exercises \(67-70,\) you will explore graphically the general sine function $$f(x)=37 \sin \left(\frac{2 \pi}{365}(x-101)\right)+25$$ as you change the values of the constants \(A, B, C,\) and \(D .\) Use a CAS or computer grapher to perform the steps in the exercises. The period \(B\) Set the constants \(A=3, C=D=0\) . a. Plot \(f(x)\) for the values \(B=1,3,2 \pi, 5 \pi\) over the interval \(-4 \pi \leq x \leq 4 \pi\) . Describe what happens to the graph of the general sine function as the period increases. b. What happens to the graph for negative values of \(B ?\) Try it with \(B=-3\) and \(B=-2 \pi\)

Find the function values in Exercises \(47-50\) . $$ \sin ^{2} \frac{\pi}{12} $$
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