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Chapter 5: Problem 43

Determine whether each inequality is true or false. $$38+76>109$$

Short Answer

Expert verified
The inequality is true.

Step by step solution

01

Evaluate the Left Side

Calculate the sum of the numbers on the left side of the inequality: \(38 + 76 = 114\)
02

Compare with the Right Side

Now that we have the sum as 114, compare it to the number on the right side of the inequality, which is 109. Since 114 is greater than 109, the inequality \(114 > 109\) holds true.
03

Conclusion

Based on the evaluation, determine the truth value of the inequality. Since 114 is greater than 109, the inequality \(38 + 76 > 109\) is true.

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Key Concepts

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

Comparison in Inequalities
Understanding comparison is essential when dealing with inequalities. Inequalities involve showing the relative size or order of numbers or expressions. The main symbols used in inequalities are:
  • <: less than
  • >: greater than
  • :
    • \(x < y\): means \(x\) is smaller than \(y\)
    • \(x > y\): means \(x\) is larger than \(y\)
    • \(x \leq y\): means \(x\) is either smaller than or equal to \(y\)
    • \(x \geq y\): means \(x\) is either larger than or equal to \(y\)
    • \(x eq y\): means \(x\) is not equal to \(y\)

    In our example, the comparison involved checking whether the sum of \(38 + 76\) is greater than \(109\). By evaluating, we established the groundwork for understanding that differences in values dictate whether one side truly is greater than, less than, or equal to another.
Basic Arithmetics in Inequalities
To solve inequalities, you often start with basic arithmetic operations like addition, subtraction, multiplication, or division. Performing these correctly is vital to understanding which side of an inequality is larger or smaller.
  • **Addition**: Combining numbers into a larger total.
  • **Subtraction**: Finding the difference between numbers.
  • **Multiplication**: Calculating the total of repeated addition of a number.
  • **Division**: Splitting a number into smaller, equal parts.
For our inequality \(38 + 76 > 109\), we calculated the sum on the left side, obtaining 114. This simple arithmetic revealed that 114 is indeed greater than 109, making the inequality true.
Why is this important? Mastering these operations helps you tackle more complex mathematical problems with confidence.
Problem Solving with Inequalities
Problem solving in mathematics involves setting a strategy and logical steps to find solutions. Solving inequalities is a part of this broader skill.
When tackling inequalities:
  • **Identify** the components: Separate what needs to be worked out.
  • **Calculate** precisely: Use arithmetic to determine exact figures and operations involved.
  • **Compare** results: Look at the numbers obtained and determine their relationship.
  • **Conclude** the truth of the statement: Decide if the inequality holds based on your calculations.
In our task, we first identified the need to find the sum of \(38 + 76\). Next, we calculated this sum to be 114. Upon comparison, it was clear that 114 is definitely greater than 109.
Thus, the inequality given is true. Structuring your approach this way not only finds solutions but enhances clarity and accuracy in problem solving.

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Most popular questions from this chapter

Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain. $$8,7,15$$

Find the measures of the sides of \(\triangle E F G\) with the given vertices and classify each triangle by its sides. \(E(16,14), F(7,6), G(-5,-14)\)

Write the assumption you would make to start an indirect proof of each statement. Points \(P, Q,\) and \(R\) are collinear.

Name the corresponding congruent angles and sides for each pair of congruent triangles. $$\triangle C D G \cong \triangle R S W$$

The SAS Inequality states that the base of an isosceles triangle gets longer as the measure of the vertex angle increases. Describe the effect of changing the measure of the vertex angle on the measure of the altitude. Justify your answer.
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