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Chapter 1: Problem 4

Evaluate each expression. $$-6-|-6|$$

Short Answer

Expert verified
-12

Step by step solution

01

Evaluate the Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Thus, for \[ |-6|, \] the result is \[ 6, \] because the absolute value of \(-6\) is \(6\).
02

Substitute and Simplify

Replace the expression \(|-6|\) with its absolute value, \(6\), in the original expression. The expression becomes \(-6 - 6.\) Then perform the subtraction: \[-6 - 6 = -12.\]

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Key Concepts

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

Understanding the Number Line
The number line is a fundamental concept that offers a simple way to visually represent numbers, including positive numbers, zero, and negative numbers. Imagine a straight horizontal line with zero positioned at the center. To its right, you have positive numbers like 1, 2, 3, and so on. To its left are negative numbers such as -1, -2, -3, etc. The number line serves as a practical tool to measure distances and understand the positions of numbers relative to each other.
  • The concept of distance on the number line is crucial when we discuss absolute values.
  • For example, both -6 and 6 are six units away from zero, although in opposite directions.
This distance-from-zero idea is what absolute value measures, making it an essential starting point when working with different mathematical operations including subtraction involving negative numbers.
The Concept of Subtraction
Subtraction is one of the core arithmetic operations that involves finding the difference between numbers. Think of subtraction as taking a step backward along the number line. For example, if you're at 4 and you subtract 2, you move two steps left on the number line ending up at 2. Now consider what happens when both numbers involved are positive and when they are negative, as each scenario influences the direction in which you move.
  • When simplifying the expression with absolute value, substitution helps by replacing it with its non-negative counterpart.
  • In the example problem, we replaced \(|-6|\) with 6 and performed the subtraction \(-6 - 6\) to arrive at \(-12\).
Subtracting larger numbers from smaller numbers results in negative outcomes, which is why understanding negative numbers becomes essential.
Exploring Negative Numbers
Negative numbers are just as important as positive numbers in mathematics. They extend the spectrum of what we can calculate on the number line to the left of zero. This allows us to accommodate real-world situations such as temperatures below freezing, underwater depths, or financial debts.
  • Negative numbers follow unique rules, especially when added or subtracted.
  • When subtracting a positive from a negative, as in \(-6 - 6\), think of it as moving further in the negative direction on the number line.
Understanding their behavior lets you better grasp concepts like absolute value. Recognizing and interpreting negative results is important for accurate calculations and problem-solving in various mathematical contexts.

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Most popular questions from this chapter

Find an equation for the line that is described, and sketch the graph. Write the answer in the form \(A x+B y+C=0\). Passes through (-3,4) and is parallel to the \(x\) -axis.

You \(\%\) need to recall the following definitions and results from elementary geometry. In a triangle, a line segment drawn from a vertex to the midpoint of the opposite side is called a median. The three medians of a triangle are concurrent; that is, they intersect in a single point. This point of intersection is called the centroid of the triangle. A line segment drawn from a vertex perpendicular to the opposite side is an altitude. The three altitudes of a triangle are concurrent; the point where the altitudes intersect is the orthocenter of the triangle. This exercise illustrates the fact that the altitudes of a triangle are concurrent. Again, we'll be using \(\triangle A B C\) with vertices \(A(-4,0), B(2,0),\) and \(C(0,6) .\) Note that one of the altitudes of this triangle is just the portion of the \(y\) -axis extending from \(y=0\) to \(y=6 ;\) thus, you won't need to graph this altitude; it will already be in the picture. (a) Using paper and pencil, find the equations for the three altitudes. (Actually, you are finding equations for the lines that coincide with the altitude segments.) (b) Use a graphing utility to draw \(\triangle A B C\) along with the three altitude lines that you determined in part (a). Note that the altitudes appear to intersect in a single point. Use the graphing utility to estimate the coordinates of this point. (c) Using simultaneous equations (from intermediate algebra), find the exact coordinates of the orthocenter. Are your estimates in part (b) close to these values?

(a) Verify that the point (3,7) is on the circle $$x^{2}+y^{2}-2 x-6 y-10=0$$ (b) Find the equation of the line tangent to this circle at the point \((3,7) .\) Hint: A result from elementary geometry says that the tangent to a circle is perpendicular to the radius drawn to the point of contact.

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=-x^{2}$$

Find the standard equation of the circle tangent to the \(x\) -axis and with center \((3,5) .\) Hint: First draw a sketch.
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