Problem 37 - Calculus Single Variable

Question

Sketch the given region. \(\\{(x, y):|x|<7,|y+4|>1\\}\)

Step-by-Step Solution

Answer

The region is two vertical strips: (-7,7) with y>-3 and (-7,7) with y<-5.

Step 1: Understand the Inequality for x

Let's analyze the inequality \(|x| < 7\). This inequality tells us that the absolute value of \(x\) must be less than 7. The absolute value measures the distance from zero, so the values of \(x\) must lie between -7 and 7. Thus, \(-7 < x < 7\).

Step 2: Understand the Inequality for y

Now examine the inequality \(|y+4| > 1\). This requires the absolute value of \(y+4\) to be greater than 1. Rewriting in terms of two separate inequalities, we have \(y + 4 > 1\) or \(y + 4 < -1\). Simplifying these gives two conditions: \(y > -3\) or \(y < -5\).

Step 3: Combine Conditions for x and y

From Step 1, we have \(-7 < x < 7\). From Step 2, we have two conditions \(y > -3\) or \(y < -5\). The region can be described by combining the \(x\) and \(y\) inequalities: \(-7 < x < 7\) and \(y > -3\) or \(-7 < x < 7\) and \(y < -5\).

Step 4: Sketch the Region

Draw the coordinate plane with the x and y axes. Shade the vertical strip \(-7 < x < 7\). Next, shade the horizontal region above \(y = -3\) line (open at \(y = -3\)) and below \(y = -5\) line (open at \(y = -5\)). The region consists of two parts: a strip going upwards from \(y = -3\) and another going downwards from \(y = -5\), both bounded by the lines \(x = -7\) and \(x = 7\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities can seem tricky at first, but they are actually quite straightforward once you understand them. Think of absolute value as a measure of distance from zero on the number line. For example, in the inequality \(|x| < 7\), we are looking at values of \(x\) whose distance from zero is less than 7.
To solve this, consider the two scenarios:

• The distance of \(x\) from zero is less than 7, so \(-7 < x < 7\).

Similarly, for the inequality \(|y+4| > 1\), you need to figure out when the distance of \(y + 4\) from zero is more than 1. This boils down to two separate conditions:

• \(y + 4 > 1\)
• \(y + 4 < -1\)

By solving these, you get \(y > -3\) or \(y < -5\). Each of these represents a separate range that \(y\) can be in.

Coordinate Plane Sketching

When sketching on the coordinate plane, the goal is to visually represent regions defined by inequalities. Let's break it down more simply with the example: \(\{(x, y):|x|<7,|y+4|>1\}\).

• Start by sketching the region defined by \(-7 < x < 7\): this is a vertical strip between the lines \(x = -7\) and \(x = 7\).
• Now consider the y-conditions: because we have two inequalities, this creates two separate regions. One region is above the line \(y = -3\), and the other is below the line \(y = -5\).
This means, you shade:
• A vertical strip above \(y = -3\)
• Another vertical strip below \(y = -5\)
Beware of boundaries: use dashed lines to indicate lines that are not actually part of the region, but serve as boundaries.

Set Notation

Set notation is a compact way to express and describe collections of numbers that satisfy certain conditions. In math, you will often see something like \(\{(x, y):|x|<7,|y+4|>1\}\) which essentially describes a set of points (x, y) that meet two criteria.
Here's how it works:

• The curly braces \(\{\}\) tell us that we're dealing with a set.
• The colon ":" reads as "such that," which leads into the conditions \(|x| < 7\) and \(|y + 4| > 1\).
• These conditions act as rules that every point \((x, y)\) in the set must satisfy.

This set notation is a powerful tool in mathematics because it allows you to quickly and efficiently communicate which numbers or points are included.

Inequalities

Inequalities express a relationship between two expressions by showing how they compare in size. There are a few key symbols to remember:

• "<" and ">" signify "less than" and "greater than", respectively.
• "\(\leq\)" and "\(\geq\)" are used for "less than or equal to" and "greater than or equal to".

When dealing with inequalities in equations, we are often finding a range or region of values that make the inequality true. For example, \(-7 < x < 7\) defines a range of \(x\) values between \(-7\) and \(7\), making it a compound inequality.

For compound inequalities, everything inside the combined range satisfies the inequality. Inequalities can be open-ended, meaning they do not include the boundary values, or closed if they do include their boundaries.
This distinction is crucial when sketching or solving inequalities.