Problem 60 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ |x|<5 $$

Step-by-Step Solution

Answer

The solution to the inequality \(|x|<5\) is \(-5

Step 1: Understand the Absolute Value Inequality

Absolute value is a measure of the magnitude of the number, regardless of its sign. An absolute value inequality like \(|x|<5\) represents a range of possible values for x where the magnitude or 'absolute' value of x is less than 5, which translates to the solution lying within the interval (-5, 5).

Step 2: Set up the Inequality

This inequality can actually be broken up into two separate inequalities because of the absolute value. This is a unique property of absolute value inequalities. So, we will really need to solve two expressions: \(x<5\) and \(-x<5\) .

Step 3: Solve the Inequalities

Now, we can separately solve these inequalities. First, \(x<5\) is already solved. For the second inequality \(-x<5\), we can multiply both sides by -1 to solve for x. Just remember that when you multiply or divide both sides of an inequality by a negative number, you need to flip the direction of the inequality symbol, so \(-x<5\) becomes \(x>-5\).

Step 4: Combine Solutions

The final solution set is the values of x that satisfy both \(x<5\) and \(x>-5\), which is \(-5

Key Concepts

Solving Inequalities

To solve inequalities means to find all the values of the variable that make the inequality true. These values are often represented as a range or as specific points on the number line. There are several steps involved in solving inequalities. First, you need to understand the inequality sign and what it's asking for. Whether it's '<', meaning less than, or '>', meaning greater than, the solution will be a set of values that satisfy the condition. It's also important to look at whether the inequality includes equals ('≤' or '≥'), which would mean the solution could also be equal to a certain number.

When encountering absolute value inequalities, such as \( |x|<5 \), it's necessary to consider both the positive and negative scenarios since absolute value indicates the distance from zero, disregarding the sign. For example, in the given inequality, we have two conditions to satisfy: \( x<5 \) and \( x>-5 \) (which is derived from \( -x<5 \) by multiplying both sides by -1 and flipping the inequality sign). The final step is to combine these inequalities to form the full solution. In this case, the values of x that satisfy the inequality lie between -5 and 5 but do not include 5 or -5 themselves.

Absolute Value

Absolute value of a number is a concept used to describe the distance of that number from zero on the number line, regardless of the direction. It is always non-negative, as it represents a magnitude without considering the sign. For a real number x, the absolute value is denoted as \( |x| \).

For understanding absolute value inequalities like \( |x|<5 \), imagine you're measuring the distance from zero on both sides of the number line. This means we're looking for all numbers whose distance from zero is less than 5 units, leading us to consider both the negative and positive directions from zero. Therefore, \( |x|<5 \) simplifies to two separate inequalities, as the value of x could be either negative or positive, but in either case, it should be within 5 units of zero in terms of magnitude. This creates a range of solutions rather than a singular answer.

Interval Notation

Interval notation is a mathematical shorthand for expressing a range of numbers, usually the solution set for inequalities. It's a compact way to display intervals on the real number line, indicating which numbers are included and which are not. In the notation, '(' or ')' denote that the endpoint is not included in the set, also known as an open interval, while '[' or ']' denote that the endpoint is included, known as a closed interval.

For the inequality \( |x|<5 \), the solution in interval notation is \( (-5, 5) \), signifying all the numbers greater than -5 and less than 5—but not -5 or 5 themselves—are included in the solution set. This notation is favored for its clarity and efficiency, especially when working with larger sets of numbers or multiple ranges.

Inequality Properties

Inequalities have unique properties that govern how they can be manipulated and solved. One fundamental property is that you can add or subtract the same number from both sides without changing the direction of the inequality. For example, if we have \( x + 2 < 7 \), subtracting 2 from both sides retains the inequality: \( x < 5 \).

However, multiplication or division by a negative number requires a reversal of the inequality sign. This is because negative numbers have an inverse effect on the direction of the inequality. For instance, when we multiply both sides of the inequality \( -x<5 \) by -1 to solve for x, the inequality reverses, becoming \( x>-5 \). In addition to these, it's vital to consider the transitive property, which tells us that if \( a

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Problem 59

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Problem 60

Other exercises in this chapter

Problem 59

Find all values of \(x\) such that \(y=0\) $$y=\frac{x+6}{3 x-12}-\frac{5}{x-4}-\frac{2}{3}$$

View solution

Problem 59

Explain how to add complex numbers. Provide an example with your explanation.

View solution

Problem 60

Solve equation by completing the square. $$ 2 x^{2}+5 x-3=0 $$

View solution

Problem 60

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$\left(y-\frac{10}{y}\right)^{2}+6\left(y-\frac{10}{y}\right)-27=0$$

View solution

Key Concepts

Solving Inequalities

To solve inequalities means to find all the values of the variable that make the inequality true. These values are often represented as a range or as specific points on the number line. There are several steps involved in solving inequalities. First, you need to understand the inequality sign and what it's asking for. Whether it's '<', meaning less than, or '>', meaning greater than, the solution will be a set of values that satisfy the condition. It's also important to look at whether the inequality includes equals ('≤' or '≥'), which would mean the solution could also be equal to a certain number.

When encountering absolute value inequalities, such as \( |x|<5 \), it's necessary to consider both the positive and negative scenarios since absolute value indicates the distance from zero, disregarding the sign. For example, in the given inequality, we have two conditions to satisfy: \( x<5 \) and \( x>-5 \) (which is derived from \( -x<5 \) by multiplying both sides by -1 and flipping the inequality sign). The final step is to combine these inequalities to form the full solution. In this case, the values of x that satisfy the inequality lie between -5 and 5 but do not include 5 or -5 themselves.

Absolute Value

Absolute value of a number is a concept used to describe the distance of that number from zero on the number line, regardless of the direction. It is always non-negative, as it represents a magnitude without considering the sign. For a real number x, the absolute value is denoted as \( |x| \).

For understanding absolute value inequalities like \( |x|<5 \), imagine you're measuring the distance from zero on both sides of the number line. This means we're looking for all numbers whose distance from zero is less than 5 units, leading us to consider both the negative and positive directions from zero. Therefore, \( |x|<5 \) simplifies to two separate inequalities, as the value of x could be either negative or positive, but in either case, it should be within 5 units of zero in terms of magnitude. This creates a range of solutions rather than a singular answer.

Interval Notation

Interval notation is a mathematical shorthand for expressing a range of numbers, usually the solution set for inequalities. It's a compact way to display intervals on the real number line, indicating which numbers are included and which are not. In the notation, '(' or ')' denote that the endpoint is not included in the set, also known as an open interval, while '[' or ']' denote that the endpoint is included, known as a closed interval.

For the inequality \( |x|<5 \), the solution in interval notation is \( (-5, 5) \), signifying all the numbers greater than -5 and less than 5—but not -5 or 5 themselves—are included in the solution set. This notation is favored for its clarity and efficiency, especially when working with larger sets of numbers or multiple ranges.

Inequality Properties

Inequalities have unique properties that govern how they can be manipulated and solved. One fundamental property is that you can add or subtract the same number from both sides without changing the direction of the inequality. For example, if we have \( x + 2 < 7 \), subtracting 2 from both sides retains the inequality: \( x < 5 \).

However, multiplication or division by a negative number requires a reversal of the inequality sign. This is because negative numbers have an inverse effect on the direction of the inequality. For instance, when we multiply both sides of the inequality \( -x<5 \) by -1 to solve for x, the inequality reverses, becoming \( x>-5 \). In addition to these, it's vital to consider the transitive property, which tells us that if \( a

Previous

Problem 59

Next

Problem 60

Other exercises in this chapter

Problem 59

Find all values of \(x\) such that \(y=0\) $$y=\frac{x+6}{3 x-12}-\frac{5}{x-4}-\frac{2}{3}$$

View solution

Problem 59

Explain how to add complex numbers. Provide an example with your explanation.

View solution

Problem 60

Solve equation by completing the square. $$ 2 x^{2}+5 x-3=0 $$

View solution

Problem 60

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$\left(y-\frac{10}{y}\right)^{2}+6\left(y-\frac{10}{y}\right)-27=0$$

View solution