Problem 29

Question

Rewrite the expression without using the absolute value symbol, and simplify the result. $$|a-b| \text { if } a

Step-by-Step Solution

Verified

Answer

The expression simplifies to $ b-a $ if $ a < b $.

1Step 1: Understand the Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, and it is always non-negative. For two numbers $ a $ and $ b $, the expression $ |a-b| $ essentially means the non-negative difference between $ a $ and $ b $.

2Step 2: Analyze the Condition a<b

Given the condition $ a < b $, the expression $ a-b $ becomes negative because $ a $ is less than $ b $. Therefore, $ |a-b| $ is the same as $ -(a-b) $ to make the result positive.

3Step 3: Simplify the Expression

Rewrite the expression $ |a-b| $ as $ -(a-b) $ when $ a < b $. Simplifying further, this is equal to $ b-a $.

4Step 4: Conclusion

Thus, the expression $ |a-b| $ under the condition $ a < b $ simplifies to $ b-a $.

Key Concepts

InequalitiesSimplifying ExpressionsDistance on Number Line

Inequalities

Inequalities are an important concept in mathematics, helping us describe the relative size of different values. When we talk about inequalities, we are essentially comparing two numbers to see which one is larger or smaller. For example, when we say that $ a < b $, it means that the number $ a $ is less than the number $ b $.

Inequalities can be represented using different symbols: less than ($<$), greater than ($>$), less than or equal to ($\leq$), and greater than or equal to ($\geq$).
They aren’t just limited to two values; inequalities can describe ranges, such as $ x > 3 $ which means any number greater than 3.

These comparisons are useful in various areas of life and mathematics because they allow us to set boundaries and conditions. In the problem here, knowing whether $ a $ is less than $ b $ helps us determine how the absolute value expression, $|a-b|$, should be handled.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This means rewriting the expression so that it's as straightforward and clear as possible. The process helps in making expressions easier to work with.

This could involve combining like terms, which are terms with the same variables raised to the same power.
Another common simplification involves removing parentheses or absolute value symbols, like in the problem $|a-b|$.
Understanding how operations like addition, subtraction, multiplication, and division affect an expression is crucial in simplifying it effectively.

In the context of our problem, simplifying $|a-b|$ when \(a

Distance on Number Line

Understanding distance on a number line is crucial when dealing with absolute values. Absolute value relates directly to how far a number is from zero, regardless of direction.

Just like on a real line, every number has a specific location, and the difference between two numbers represents how far apart they are.
This difference is always a positive value since distance cannot be negative--hence why absolute values are non-negative.
When we say $|a-b|$, we are actually calculating how far apart numbers $a$ and $b$ are on a number line.

In this problem, the condition $a < b$ means that $a$ is left of $b$ on the number line. Therefore, the expression transforms to $b-a$ because it represents the positive distance between the two points, illustrating the distance as if they were flipped to ensure positivity without direct regard to order.

Problem 29

Other exercises in this chapter

Problem 28

Write the expression in the form $a+b i,$ where $a$ and $b$ are real numbers. $$(3-2 i)^{3}$$

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Problem 29

Solve the equation. $9 x^{3}-18 x^{2}-4 x+8=0$

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Problem 29

Simplify. $$\left(5 x^{2} y^{-3}\right)\left(4 x^{-5} y^{4}\right)$$

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Problem 29

Factor the polynomial. $$x^{2}-49 y^{2}-14 x+49$$

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