/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 126 The slopes of two lines are \(-4... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 126

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

Short Answer

Expert verified
The line with the slope of -4 is steeper as its absolute value is higher than that of the other line.

Step by step solution

01

Understand the concept of slope

The slope of a line, often represented by the letter 'm', is a measure that defines the steepness, incline, or gradient of the line. A higher absolute value of the slope signifies a steeper incline.
02

Convert fractions to decimals

While comparing, it's easier to have all the numbers in the same format. The fraction 5/2 converts to 2.5 in decimal format.
03

Compare the absolute slopes

Take the absolute value of both slopes. The absolute value of -4 is 4, and the absolute value of 2.5 is 2.5. Now it's easier to see that 4 is higher than 2.5.

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Key Concepts

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

Comparing Slopes
When we talk about comparing slopes in mathematics, we are essentially trying to determine which line on a graph is steeper. The slope, designated usually as "m", helps us understand how fast a line rises or falls as we move from left to right across a graph.

A larger absolute value of a slope means the line is steeper. For example, if you have slopes like y = -4x and y = \(\frac{5}{2}x\), looking at their absolute values鈥�4 and 2.5 respectively鈥攜ou can tell which line is steeper by picking the greater absolute slope value. Here, the line with a slope of -4 is steeper, because its absolute value is 4. Comparing slopes in this manner is key to understanding relationships between lines and how their inclinations differ.
Absolute Value
Understanding the concept of absolute value is crucial not just in slope comparison but throughout mathematics. The absolute value of a number is its distance from zero on a number line, regardless of its direction. This means that both -4 and 4 have an absolute value of 4.

In practical terms, you disregard the negative sign when computing the absolute value. This concept is particularly useful when you want to compare negative and positive slopes, as it normalizes the values making comparison straightforward. While the slope -4 is negative, its absolute value allows us to compare it directly with the positive slope of \(\frac{5}{2}\) by highlighting their 'steepness' without concern for directionality. This ensures clarity when discussing how steep a line truly is, focusing only on magnitude.
Converting Fractions to Decimals
Converting fractions to decimals can simplify many mathematical tasks, especially when comparisons are necessary. To convert a fraction like \(\frac{5}{2}\) to a decimal, you simply perform the division: 5 divided by 2 equals 2.5.

Decimals offer a more direct format for understanding sizes and values at a glance, making it easier to compare different numbers, such as slopes or measurements. By converting the fraction \(\frac{5}{2}\) into 2.5, we make it easier to compare it with other numeric forms, such as -4.
  • Fractions can be awkward to compare at times because of differing numerators and denominators.
  • Decimals remove these extra steps, offering straightforward comparison.
  • This is particularly important in real-life applications, like calculating angles and slopes in engineering or physics.

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Most popular questions from this chapter

In Exercises 27 and 28, use the table of values for \(y=f(x)\) to complete a table for \(y=f^{-1}(x)\). $$ \begin{array}{|l|r|r|r|r|r|r|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & -10 & -7 & -4 & -1 & 2 & 5 \\ \hline \end{array} $$

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from ground level at a velocity of 96 feet per second. $$ t_{1}=2, t_{2}=5 $$

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. If the inverse function of \(f\) exists and the graph of \(f\) has a \(y\)-intercept, the \(y\)-intercept of \(f\) is an \(x\)-intercept of \(f^{-1}\).

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from a height of 6 feet at a velocity of 64 feet per second. $$ t_{1}=0, t_{2}=3 $$

The numbers of households \(f\) (in thousands) in the United States from 1995 to 2003 are shown in the table. The time (in years) is given by \(t\), with \(t=5\) corresponding to 1995 . (Source: U.S. Census Bureau) $$ \begin{array}{|c|c|} \hline \text { Year, } t & \text { Households, } f(t) \\ \hline 5 & 98,990 \\ 6 & 99,627 \\ 7 & 101,018 \\ 8 & 102,528 \\ 9 & 103,874 \\ 10 & 104,705 \\ 11 & 108,209 \\ 12 & 109,297 \\ 13 & 111,278 \\ \hline \end{array} $$ (a) Find \(f^{-1}(108,209)\). (b) What does \(f^{-1}\) mean in the context of the problem? (c) Use the regression feature of a graphing utility to find a linear model for the data, \(y=m x+b\). (Round \(m\) and \(b\) to two decimal places.) (d) Algebraically find the inverse function of the linear model in part (c). (e) Use the inverse function of the linear model you found in part (d) to approximate \(f^{-1}(117,022)\). (f) Use the inverse function of the linear model you found in part (d) to approximate \(f^{-1}(108,209)\). How does this value compare with the original data shown in the table?
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