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Chapter 5: Problem 63

Determine whether each equation is true or false if \(a=2\) \(b=5,\) and \(c=6.\) $$a+c>a+b$$

Short Answer

Expert verified
The equation is true.

Step by step solution

01

Substitute values into the equation

The original equation given is \(a+c > a+b\). Substitute \(a = 2\), \(b = 5\), and \(c = 6\) into the equation. This becomes \(2 + 6 > 2 + 5\).
02

Simplify both sides of the inequality

Calculate the sums on both sides of the inequality: \(2 + 6 = 8\) and \(2 + 5 = 7\). The inequality now becomes \(8 > 7\).
03

Determine the truth value of the inequality

Since \(8\) is indeed greater than \(7\), the inequality \(8 > 7\) is true.

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Key Concepts

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

Substitution
Substitution is a crucial first step when dealing with expressions and inequalities involving variables. This involves replacing the variables with their given numerical values to simplify the inequality.
  • In equations or inequalities, variables symbolize unknown or given quantities that need specific values for evaluation.
  • Substitution helps transform an abstract mathematical expression into something more structured and manageable.
Consider the inequality in our exercise: \(a + c > a + b\). To evaluate whether this statement is true or false with the given values of \(a = 2\), \(b = 5\), and \(c = 6\), we substitute these values into the inequality. This results in: \(2 + 6 > 2 + 5\). Substitution is a straightforward yet powerful tool used to understand and solve expressions involving variables.
Simplifying expressions
After substituting values into the inequality, the next task is to simplify the expressions on both sides. Simplification involves reducing the expressions to their simplest form by performing basic arithmetic operations such as addition or multiplication.
  • It helps in achieving clarity by breaking down complex expressions into basic components.
  • In inequalities, simplification aids in directly comparing values on either side.
For the inequality \(2 + 6 > 2 + 5\), simplifying involves adding the numbers on each side: * Left side: \(2 + 6 = 8\) * Right side: \(2 + 5 = 7\) After simplification, the inequality transforms into \(8 > 7\), making it clear that the expression on the left is greater.
Comparing values
Once we have simplified both sides of an inequality, the final step is to compare these values to determine the inequality's truth value. This involves assessing the relationship between the two results.
  • Comparison involves checking whether one value is greater than, less than, or equal to the other.
  • This determines if the expression correctly represents the mathematical scenario described.
In our case: * Left side after simplification: \(8\)* Right side after simplification: \(7\)The inequality \(8 > 7\) confirms that the original inequality \(a + c > a + b\) holds true when \(a = 2\), \(b = 5\), and \(c = 6\). Through comparison, we determine that our evaluated expression accurately describes the relationship between the substituted variables, leading us to the conclusion that the statement is indeed true.

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