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Chapter 5: Problem 41

Determine whether each inequality is true or false. $$19-10<11$$

Short Answer

Expert verified
The inequality \(19 - 10 < 11\) is true.

Step by step solution

01

Simplify the Left Side

First, perform the subtraction on the left side of the inequality. Calculate \( 19 - 10 \). This simplifies to \( 9 \).
02

Compare Values

Now, compare the result obtained (\(9\)) against the number on the right side of the inequality sign, which is \(11\). The inequality is \(9 < 11\).
03

Determine True or False

Since \(9\) is indeed less than \(11\), the inequality \(9 < 11\) holds true. Therefore, the original inequality \(19 - 10 < 11\) is true.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions involves reducing an expression to its simplest form. This means performing all possible arithmetic operations. It can also involve combining like terms or using algebraic rules. When we simplify, our goal is to make the expression as basic and straightforward as possible.

In the exercise given, we start by simplifying the expression on the left side of the inequality. We have to solve the subtraction of two numbers: calculate \(19 - 10\). Doing this calculation, we simplify it to \(9\).
  • Subtract \(10\) from \(19\)
  • The result is \(9\)
By simplifying, we can then work with simpler numbers, making it easier to proceed to further steps such as comparison.
Comparison of Numbers
After simplifying expressions, the next step is the comparison of numbers. This is where we determine the relationship between two values. The signs we use for comparison can be greater than \(>\), less than \(<\), or equal to \(=\). Knowing how to interpret these signs is crucial in understanding inequalities.

Here, we take our simplified number, \(9\), and compare it with something else, which is \(11\) in this instance. We look at the inequality sign \(<\) to guide us. We know that \(9 < 11\) means that \(9\) is indeed less than \(11\).
  • Compare \(9\) with \(11\)
  • Check if \(9\) is less than \(11\)
Since the statement is true, this comparison helps us determine the validity of the original inequality expression.
True or False Statements
A true or false statement asks us to verify the accuracy of a claim. In mathematics, especially inequalities, determining whether something is true or false involves comparing numbers or expressions logically.

For the original expression \(19 - 10 < 11\), once simplified and compared, the final step is to decide its truth value.
  • Is \(9 < 11\)?
  • Yes, since \(9\) is less than \(11\)
Therefore, the original inequality statement is true. True or false evaluations help to validate our calculations and ensure the initial expressions are correctly interpreted. This is essential in problem-solving and checking our work efficiency.

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Most popular questions from this chapter

Name the corresponding congruent angles and sides for each pair of congruent triangles. $$\triangle T U V \cong \triangle X Y Z$$

List the sides of \(\triangle P Q R\) in order from longest to shortest if the angles of \(\triangle P Q R\) have the given measures. \(m \angle P=7 x+8, m \angle Q=8 x-10, m \angle R=7 x+6\)

Find the value of \(n .\) List the sides of \(\triangle P Q R\) in order from shortest to longest for the given angle measures. $$m \angle P=9 n+29, m \angle Q=93-5 n, m \angle R=10 n+2$$

Draw \(\triangle A B C\) such that \(m \angle A>m \angle B>m \angle C .\) Do not measure the angles. Explain how you know the greatest and least angle measures.

Theorem: Angles supplementary to the same angle are congruent. Dia is proving the theorem above by contradiction. She began by assuming that \(\angle A\) and \(\angle B\) are supplementary to \(\angle C\) and \(\angle A \not \equiv \angle B\) Which of the following reasons will Dia use to reach a contradiction? \(A\) If two angles form a linear pair, then they are supplementary angles. \(B\) If two supplementary angles are equal, the angles each measure 90 . \(C\) The sum of the measures of the angles in a triangle is \(180 .\) \(D\) If two angles are supplementary, the sum of their measures is \(180 .\)
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