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Chapter 1: Problem 8

Rewrite without using the absolute value symbol. $$ |7+x| \text { if } x \geq-7 $$

Short Answer

Expert verified
|7 + x| simplifies to 7 + x when x 鈮� -7.

Step by step solution

01

Understanding the Absolute Value

The expression \(|7 + x|\) denotes the absolute value of \(7+x\). The absolute value of a number is its distance from zero on the number line, regardless of direction, which results in a non-negative value.
02

Identifying the Domain Condition

We have the condition \(x \geq -7\). This condition suggests that the value inside the absolute value, \(7 + x\), will be zero or positive since \(x\) is at least \(-7\).
03

Applying the Absolute Value Rule for Non-negative Values

For any expression \(|a|\), where \(a\) is non-negative, \(|a| = a\). Since \(7 + x\) is non-negative when \(x \geq -7\), we can write \(|7+x| = 7+x\).
04

Conclusion

Therefore, when \(x \geq -7\), the expression \(|7+x|\) can be rewritten as \(7+x\) without the absolute value symbol.

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Key Concepts

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

Non-Negative Value
The concept of a non-negative value is central to understanding absolute value. Absolute value is about the distance of a number from zero, meaning it is always a non-negative value. Whether you're dealing with positive numbers or zero, the outcome is always zero or positive.
For example, the absolute value of both 5 and -5 is 5. This is because both numbers are 5 units away from zero on the number line:
  • The absolute value of a positive number remains unchanged because it is already non-negative.
  • For negative numbers, the absolute value negates the negative sign, turning it into a positive number.
This property simplifies expressions by ensuring they yield non-negative results.
Number Line
The number line is a powerful tool for visualizing absolute value. It allows us to see the position of numbers in relation to zero, which is key to understanding the distance concept of absolute value. A number line stretches horizontally, with zero right in the middle. Numbers to the right are positive, while numbers to the left are negative.
This visualization:
  • Helps gauge the distance of numbers from zero.
  • Illustrates how absolute value "flattens" numbers to the right, making them non-negative. This means even if the position is on the left side (negative), in terms of distance, it can be expressed as a non-negative number.
Understanding this line aids in grasping concepts like the non-negative value inherently present in absolute values.
Domain Condition
Domain conditions are restrictions that specify what values a variable can take in an equation. They play a crucial role in determining the context within which you can simplify expressions like absolute values.In our example, the condition is given as:
  • \(x \geq -7\)
This means that the expression inside the absolute value,
  • \(7 + x\), is either zero or positive when \(x\) is greater than or equal to -7.
Knowing the domain condition allows us to simplify the absolute value without changing its meaning since it confirms that the expression is non-negative.
Algebraic Expression
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Simplifying these expressions鈥攕uch as removing the absolute value sign when feasible鈥攈elps in algebraic problem-solving.In the example, you start with
  • \(|7+x|\)
Given the domain condition
  • \(x \geq -7\)
We know that
  • \(7 + x\) is non-negative, allowing us to rewrite \(|7+x| = 7+x\)
In doing so, you transform the original expression into a straightforward algebraic expression. Simplification is crucial in algebra to ensure expressions are in their simplest form for further calculations or problem-solving.

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