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Chapter 1: Problem 4

The solution set of the inequality \(|x|<5\) is \({x|______________}.\)

Short Answer

Expert verified
{x | -5 < x < 5}

Step by step solution

01

Understand the Absolute Value Inequality

The inequality \( |x| < 5 \) signifies that the absolute value of x is less than 5. This means that x is restricted to a range of values that are within 5 units of 0 on the number line.
02

Break Down the Inequality

An absolute value inequality of the form \( |x| < a \) (where a is a positive number) can be broken down into two separate inequalities: \( -a < x < a \). For this problem, it translates to: \(-5 < x < 5\).
03

State the Solution Set

Based on breaking down the absolute value inequality, the solution set for \( |x| < 5 \) is the set of all x values between -5 and 5 (excluding -5 and 5). This is written in set-builder notation as \{x | -5 < x < 5 \}.

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Key Concepts

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

absolute value
Absolute value measures the distance of a number from zero on the number line, regardless of direction. It is always positive or zero. For example, the absolute value of both 3 and -3 is 3, represented as \(|3| = 3\) and \(|-3| = 3\). To visualize, imagine how far a number is from zero, ignoring whether it's to the left or right.
In inequalities, absolute value tells us how far a number can be from zero, setting boundaries in both positive and negative directions.
inequality
An inequality compares two values, showing whether one is less than, greater than, or equal to the other. In the context of absolute value inequalities, we are dealing with ranges. For instance, \(|x| < 5\) indicates that the distance of x from zero must be less than 5 units. To solve such inequalities, we split them into two separate inequalities. For \(|x| < 5\), it can be broken down into \(-5 < x < 5\).
This means x can be any number between -5 and 5, making our solution a continuous range.
set-builder notation
Set-builder notation is a concise way of expressing a set of solutions for inequalities. Instead of listing all possible values, we use a compact form: \({x | -5 < x < 5}\). This notation tells us that our solution includes all x values that satisfy the condition \(-5 < x < 5\). It's a neat way to specify large sets, especially when they include ranges or particular conditions.
number line
A number line is a visual tool to represent numbers and inequalities. It helps you see the range of solutions. To show \(|x| < 5\), you would draw a line from -5 to 5, not including -5 and 5 themselves. You can use open circles at -5 and 5 to indicate that these points are not part of the solution set.
By using a number line, you can easily visualize and understand where the solution set lies, making abstract concepts more concrete and easier to grasp.

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Most popular questions from this chapter

Find the real solutions, if any, of each equation. $$ \sqrt[4]{4-3 x^{2}}=x $$

Find the real solutions, if any, of each equation. $$ 3 x^{2}+7 x-20=0 $$

The distance to the surface of the water in a well can sometimes be found by dropping an object into the well and measuring the time elapsed until a sound is heard. If \(t_{1}\) is the time (measured in seconds) that it takes for the object to strike the water, then \(t_{1}\) will obey the equation \(s=16 t_{1}^{2}\), where \(s\) is the distance (measured in feet). It follows that \(t_{1}=\frac{\sqrt{s}}{4}\). Suppose that \(t_{2}\) is the time that it takes for the sound of the impact to reach your ears. Because sound waves are known to travel at a speed of approximately 1100 feet per second, the time \(t_{2}\) to travel the distance \(s\) will be \(t_{2}=\frac{s}{1100} .\) See the illustration. Now \(t_{1}+t_{2}\) is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard. We have the equation $$ \text { Total time elapsed }=\frac{\sqrt{s}}{4}+\frac{s}{1100} $$ Find the distance to the water's surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.

Find the real solutions, if any, of each equation. $$ 4(w-3)=w+3 $$

Find all the complex solutions of \(z^{6}+28 z^{3}+27=0\).
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