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Chapter 1: Problem 30

State whether each inequality is true or false. (a) \(-6<-10\) (b) \(\sqrt{2}>1.41\)

Short Answer

Expert verified
(a) False, (b) True

Step by step solution

01

Evaluate Inequality (a)

Examine the inequality \(-6<-10\). When evaluating this, consider that \(-6\) is greater than \(-10\) because \(-10\) is more to the left on the number line compared to \(-6\). Therefore, the statement is False.
02

Evaluate Inequality (b)

Consider the inequality \( \sqrt{2} > 1.41 \). First, determine the approximate value of \( \sqrt{2} \), which is about \( 1.414 \). Since \( 1.414 \) is greater than \( 1.41 \), the statement is True.

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Key Concepts

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

Greater Than
In mathematics, the concept of "greater than" is a fundamental idea used to compare numbers or quantities. We use the symbol ">" to indicate that one number is larger than another. For example, in the inequality \(-6 > -10\), \(-6\) is greater than \(-10\). How can this be if they're both negative?
  • Consider the number line: numbers increase in value as you move from left to right.
  • Even though \(-6\) and \(-10\) are both negative, \(-6\) is closer to zero. Therefore, it is bigger.

It's like comparing debts: owing someone \(-6\) dollars is better (greater) than owing them \(-10\). When you look at inequalities, position on the number line plays a big role.
Always check which number is further to the right when figuring out which is greater. This simple rule helps decide which inequality is true.
Less Than
The "less than" symbol "<" is used to show that one value is smaller than another. When comparing negative numbers, it might appear a little counter-intuitive at first glance.Take the inequality \(-6 < -10\). Is it true or false?
  • Look at a number line. \(-10\) is to the left of \(-6\), which means \(-10\) is actually smaller.
  • Thus, the statement \(-6 < -10\) is false, because \(-10\) is less, or more negative, than \(-6\).

Remember, in number comparisons, particularly involving negatives, the more left a number sits on the number line, the smaller it is. Use this principle whenever comparing numbers with "less than." This understanding is essential to prevent mistakes, especially when dealing with negatives.
Square Root Approximation
Understanding the concept of a square root and its approximation is crucial in math. The square root of a number \(x\) is a value that, when multiplied by itself, will give \(x\).For example, consider \(\sqrt{2} > 1.41\).
  • We need to find the approximate value of \(\sqrt{2}\). It's around \(1.414\).
  • Since \(1.414\) is larger than \(1.41\), the inequality stands true.

Using approximations like \(1.414\) for \(\sqrt{2}\) helps quickly assess inequalities without a calculator.
When dealing with square roots, knowing approximations aids in comparing values easily. Remember, knowing key square root values, such as \(\sqrt{2}\) and \(\sqrt{3}\), allows for swift mental math and enhances problem-solving efficiency.

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Most popular questions from this chapter

State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$\frac{x}{x+y}=\frac{1}{1+y}$$

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$2 x^{1 / 3}(x-2)^{2 / 3}-5 x^{4 / 3}(x-2)^{-1 / 3}$$

Plot the points \(M(6,8)\) and \(A(2,3)\) on a coordinate plane. If \(M\) is the midpoint of the line segment \(A B,\) find the coordinates of \(B .\) Write a brief description of the steps you took to find \(B\), and your reasons for taking them.

State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$2\left(\frac{a}{b}\right)=\frac{2 a}{2 b}$$

Factor the expression completely. $$x^{3}+3 x^{2}-x-3$$
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