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Chapter 4: Problem 104

From the definition of absolute value, \(|x|=x\) only for \(x \geq 0 .\) Solve \(|3 t-5|=3 t-5\) using this same reasoning.

Short Answer

Expert verified
The equation \(|3t-5|=3t-5\) holds true for \(t \geq \dfrac{5}{3}\).

Step by step solution

01

Understand Absolute Value Definition

Recall that \(|x|=x\) only for \(x \geq 0\). This means the expression inside the absolute value must be greater than or equal to zero for the equation to hold true.
02

Set Up the Inequality

For the given \(|3t-5|=3t-5\), set the expression inside the absolute value, \(3t - 5\), greater than or equal to zero: \(3t - 5 \geq 0\).
03

Solve the Inequality

Solve the inequality \(3t - 5 \geq 0\):\[\begin{align*}3t - 5 &\geq 0\quad\text{(Add \(5\) to both sides)}\3t &\geq 5\quad\text{(Divide by \(3\))}\t &\geq \dfrac{5}{3}\end{align*}\]
04

Conclusion

Thus, \(|3t-5|=3t-5\) holds true when \(t \geq \dfrac{5}{3}\).

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Key Concepts

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

inequality
Inequalities are mathematical expressions that show the relationship between two values where they are not necessarily equal. They use symbols like \(>\), \(<\), \(\geq\) (greater than or equal to), and \(\leq\) (less than or equal to). In our problem, we need to establish the condition where the expression inside the absolute value is greater than or equal to zero. This is important because for the absolute value equation to hold true, the expression inside must satisfy this inequality. For our exercise, we set up the inequality as follows:
  • Given: \(|3t-5|=3t-5\)
  • Set up the inequality: \(3t - 5 \geq 0\)
Solving this inequality helps us find the range of values for which the given absolute value expression holds true.
absolute value
The absolute value of a number is its distance from zero on the number line, without considering direction. It is always a non-negative value. The notation for absolute value is vertical bars around the number or expression, like this: \(|x|\). By definition, \(|x|=x\) if \(x \geq 0\), and \(|x|=-x\) if \(x < 0\). In the exercise given, we use the fact that the expression inside the absolute value remains the same only when it is non-negative. This concept leads to the creation of our inequality:
  • Consider \(|3t-5|\)
  • According to the definition, \(3t-5\) must be non-negative
So, we solve for when \(3t-5 \geq 0\) to maintain the same form of the expression.
solving inequalities
Solving inequalities involves finding the range of values that satisfy the inequality condition. Here鈥檚 how we solve the inequality from our exercise step-by-step:
  • Start with the inequality: \(3t - 5 \geq 0\)
  • Add 5 to both sides: \(3t \geq 5\)
  • Divide by 3: \(t \geq \frac{5}{3}\)
When you solve inequalities, it鈥檚 essential to perform the same operations on both sides of the inequality to maintain its balance. This will help in isolating the variable and finding the precise range of values that solve the inequality.
algebraic expressions
Algebraic expressions consist of numbers, variables, and arithmetic operations. They form the backbone of many mathematical problems, including inequalities. In our exercise, we deal with an algebraic expression within the absolute value notation. Understanding how to manipulate and interpret these expressions is crucial. Here's a breakdown:
  • Expression inside absolute value: \(3t-5\)
  • This needs to be made non-negative: \(3t-5 \geq 0\)
By isolating the variable \(t\), we determine the values that make this expression valid, leading to our solution. Working through these steps strengthens our understanding of managing algebraic expressions in the context of inequalities and absolute values.

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