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Chapter 0: Problem 64

Show that if \(a

Short Answer

Expert verified
To show that if \(a<b\) and \(c \leq d\), then \(a+c<b+d\), we add the given inequalities side by side: \(a < b\) and \(c \leq d\). Then, we add the left sides (\(a + c\)) and right sides (\(b + d\)) of both inequalities. Since the sum of two inequalities maintains the inequality if the signs have the same direction, we can write the inequality for the sum: \(a + c < b + d\). Therefore, we have shown that if \(a < b\) and \(c \leq d\), then \(a+c < b+d\).

Step by step solution

01

Identify the given inequalities and relationships

We are given two inequalities: 1. \(a < b\) 2. \(c \leq d\) Our goal is to show that if these inequalities hold, then \(a + c < b + d\).
02

Add inequalities

We start by adding the given inequalities together. First, we add inequality 1 to inequality 2, side by side: \(a < b\) and \(c \leq d\) Now, add the left sides of both inequalities: \(a + c\) Similarly, add the right sides of both inequalities: \(b + d\)
03

Apply properties of inequality

Since the sum of two inequalities maintains the inequality if the signs have the same direction, we can write the inequality for the sum: \(a + c < b + d\)
04

Conclude the solution

By adding the given inequalities and using the properties of inequalities, we've shown that if \(a < b\) and \(c \leq d\), then \(a+c < b+d\). This completes the proof.

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Key Concepts

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

Adding Inequalities
In the realm of precalculus, working with inequalities often requires the combination of two or more relations. Adding inequalities is a fundamental technique used to solve a wide range of problems. When adding inequalities, it's crucial to understand that the inequalities must 'face' the same way.

For example, when given two inequalities, such as \(a < b\) and \(c \leq d\), we can add them together, as long as we keep the direction of the inequalities intact. The process involves adding the left-hand sides together and separately adding the right-hand sides. Once combined, this forms a new inequality: \(a + c < b + d\) or \(a + c \leq b + d\), depending on whether the inequalities have strict '<' or the more inclusive '\leq' symbol.

To make the concept easier to understand, imagine you have two separate pieces of string of different lengths. One represents \(a < b\), and the other represents \(c \leq d\). When you tie them together end to end, it is clear that the combined length of the small pieces (\

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