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Chapter 1: Problem 142

The slopes of two lines are \(-4\) and \(\frac{5}{2}\). Which is steeper? Explain.

Short Answer

Expert verified
The line with slope \(-4\) is steeper than the line with slope \(\frac{5}{2}\). This is because the absolute value of \(-4\) is greater than the absolute value of \(\frac{5}{2}\), meaning that the line with slope \(-4\) has a greater steepness.

Step by step solution

01

Identify the given slopes

The given slopes are \(-4\) and \(\frac{5}{2}\).
02

Compare absolute values of the slopes

The absolute value of \(-4\) is \(4\) and the absolute value of \(\frac{5}{2}\) is \(2.5\). Comparing these absolute values, it can be seen that \(4 > 2.5\). Therefore, the line with slope \(-4\) is steeper.

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Key Concepts

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

Understanding Absolute Value
Absolute value is a crucial concept when comparing the steepness of lines. The absolute value of a number is its distance from 0 on a number line, regardless of direction.

It turns every negative number into a positive one and keeps positive numbers unchanged. For example, the absolute value of
  • \(-4\) is \(4\)
  • \(\frac{5}{2}\) is \(2.5\)
In geometry, we often use absolute values to compare slopes, as they're interested in the sheer size or magnitude of the slope, not its direction. This means that when comparing two slopes, we look at their absolute values to decide which one is steeper.
Line Comparison Through Slopes
To compare two lines based on slopes, it's important to understand what the slope represents. A slope indicates how much a line rises or falls as it moves horizontally.

When you have two slopes like \(-4\) and \(\frac{5}{2}\), you need to consider their magnitude. The larger the absolute value, the steeper the line.
  • First, find the absolute value of each slope.
  • For \(-4\), the absolute value is \(4\).
  • For \(\frac{5}{2}\), the absolute value is \(2.5\).
Since \(4 > 2.5\), the line with the slope of \(-4\) is steeper.
Understanding Steepness of Lines
Steeper lines represent greater changes over the same horizontal distance. This means that a steep line rises or falls more sharply.

The steeper a line, the larger its absolute value of the slope. Steepness affects various real-world situations like roads, stairs, and roofs.
  • A slope of \(-4\) indicates a very sharp decline.
  • A slope of \(\frac{5}{2}\) is less steep and represents a more gradual incline.
Recognizing steepness is important in design and safety considerations where inclines and declines must be measured precisely.

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