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Chapter 1: Problem 33

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

Analyze the first inequality

The first inequality is \(-6 < -10\). In the number line, numbers to the right are greater than those to the left. Since \(-6\) is to the right of \(-10\), it is greater than \(-10\). Thus, the inequality \(-6 < -10\) is false.
02

Analyze the second inequality

The second inequality is \(\sqrt{2} > 1.41\). First, calculate \(\sqrt{2}\) which is approximately \(1.414\). Compare this to \(1.41\). Since \(1.414\) is greater than \(1.41\), the inequality is true.

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Key Concepts

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

Number line
The number line is a visual representation of real numbers laid out on a straight line. This line helps to easily compare the size of numbers, as numbers further to the right are larger than those to the left.
When analyzing inequalities with negative numbers, it's crucial to understand their position on the number line. For instance, with the numbers -6 and -10, you might initially think that because 10 is numerically larger than 6, -10 should be greater. However, in the context of the number line:
  • -6 is positioned to the right of -10.
  • This means -6 is greater than -10.
Thus, the inequality -6 < -10 is actually false because a smaller negative number indicates a greater value comparatively.
Square root
Understanding square roots is essential when comparing specific numeric values, particularly in inequalities. The square root of a number, symbolized as \( \sqrt{} \), represents a value which, when multiplied by itself, gives the original number. To illustrate:
The square root of 2 is often approximated as 1.414. Despite being an approximation, this value is crucial for comparing with other numbers such as 1.41.
  • When comparing 1.414 and 1.41, notice that 1.414 is slightly larger.
  • This can be checked by observing both numbers past the decimal point.
Hence, \( \sqrt{2} > 1.41 \), making the inequality true. Calculators can help with more precise values, adding to conceptual comprehension.
True or false inequalities
In the context of mathematics, inequalities describe the relative size or order of two values. An inequality can be true or false, just like the basic concepts of true and false logic. When you say an inequality is true, the left-hand side is genuinely smaller or greater (as stated) than the right-hand side. Conversely, if it's false, the opposite holds.
For debunking inequalities like \(-6 < -10\), one employs comparison skills, often using tools such as number lines. Similarly, for \( \sqrt{2} > 1.41 \), familiarity with square roots is vital.
  • Assess each side of the inequation step-by-step.
  • Acknowledge visual aids like graphs or calculators.
  • Regularly practice to build accuracy.
This consistent approach helps clarify whether an inequality is true or false, empowering a strong foundational understanding.

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