Problem 57 - College Algebra

Question

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|x|<3$$

Step-by-Step Solution

Answer

The solution to the inequality \(|x|<3\) is \(-3

Step 1: Rewrite the absolute value inequality without absolute value bars

An absolute value inequality \(|x|<3\) can be rewritten as \( -3-3\). This is because the definition of an absolute value is that it is the distance a number is from zero on the number line, which can be either positive or negative.

Step 2: Solve the inequalities

Solving the inequality \(x<3\) gives all the real numbers less than 3, and solving \(x>-3\) gives all the real numbers more than -3. Since we need to satisfy both inequalities simultaneously, the solution would be the intersection of both of these sets. Thus, the solution of the inequality \(|x|<3\) is the set of all x such that \(-3

Step 3: Graph the solution set on the number line

The solution set can be graphed on a number line by marking both -3 and 3 on the number line and shading the region between them. This is because all the values between -3 and 3 are in solution set.

Step 4: Express the solution set using interval notation

The solution set \(-3

Key Concepts

Understanding the Solution Set

The solution set of an inequality refers to the set of all possible values that satisfy the inequality. In the case of the absolute value inequality \(|x| < 3\), we find its solution set by rewriting it into two separate inequalities:

• \(x < 3\)
• \(x > -3\)

These inequalities together mean that \(x\) must be greater than \(-3\) but also less than \(3\). In simpler terms, it's the set of numbers that fit between -3 and 3 on the number line.
This forms the solution set \(-3 < x < 3\), which includes all numbers between -3 and 3 but not -3 and 3 themselves. This solution set gives us the range of permissible values for \(x\) that make the original inequality true.

Interval Notation

Interval notation is a convenient way of expressing sets of numbers, especially useful for solution sets derived from inequalities. For the inequality \(-3 < x < 3\), the solution set is expressed in interval notation as \((-3, 3)\).

• The parenthesis \((\) and \()\) indicate that -3 and 3 are not included in the set (this is called an "open interval").
• If the interval were \([-3, 3]\), it would mean -3 and 3 are included, shown by using brackets \([\) and \()\) (this is a "closed interval").

Interval notation like \((-3, 3)\) provides a concise way to represent the solution set for an inequality, indicating the start and end points of the range and whether they are included or excluded.

Graphing the Solution Set on a Number Line

The number line is a fundamental tool for visualizing the solution set. To graph the solution set \(-3 < x < 3\), we begin by marking the points -3 and 3 on the number line. However, because these points are not included in the solution set (as denoted by the open interval \((-3, 3)\)), we use open circles for -3 and 3.
Next, we shade the region between these two points, representing all the values that \(x\) can take.

• The open circles show that the endpoint numbers (in this case, -3 and 3) are not part of the solution set.
• The shaded part in between illustrates all the real numbers that satisfy the inequality \(-3 < x < 3\).

Graphing on a number line helps to visually confirm the range of solutions and provides a quick way to interpret the inequality.

Previous

Problem 57

Next

Problem 57

Other exercises in this chapter

Problem 57

Solve each equation in Exercises \(55-64\) using the quadratic formula. $$ x^{2}+5 x+3=0 $$

View solution

Problem 57

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=l w\) for \(w\)

View solution

Problem 57

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ \frac{2 x}{x-3}=\frac{6}{x-3}+4

View solution

Problem 58

Solve each equation in 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

Understanding the Solution Set

The solution set of an inequality refers to the set of all possible values that satisfy the inequality. In the case of the absolute value inequality \(|x| < 3\), we find its solution set by rewriting it into two separate inequalities:

• \(x < 3\)
• \(x > -3\)

These inequalities together mean that \(x\) must be greater than \(-3\) but also less than \(3\). In simpler terms, it's the set of numbers that fit between -3 and 3 on the number line.
This forms the solution set \(-3 < x < 3\), which includes all numbers between -3 and 3 but not -3 and 3 themselves. This solution set gives us the range of permissible values for \(x\) that make the original inequality true.

Interval Notation

Interval notation is a convenient way of expressing sets of numbers, especially useful for solution sets derived from inequalities. For the inequality \(-3 < x < 3\), the solution set is expressed in interval notation as \((-3, 3)\).

• The parenthesis \((\) and \()\) indicate that -3 and 3 are not included in the set (this is called an "open interval").
• If the interval were \([-3, 3]\), it would mean -3 and 3 are included, shown by using brackets \([\) and \()\) (this is a "closed interval").

Interval notation like \((-3, 3)\) provides a concise way to represent the solution set for an inequality, indicating the start and end points of the range and whether they are included or excluded.

Graphing the Solution Set on a Number Line

The number line is a fundamental tool for visualizing the solution set. To graph the solution set \(-3 < x < 3\), we begin by marking the points -3 and 3 on the number line. However, because these points are not included in the solution set (as denoted by the open interval \((-3, 3)\)), we use open circles for -3 and 3.
Next, we shade the region between these two points, representing all the values that \(x\) can take.

• The open circles show that the endpoint numbers (in this case, -3 and 3) are not part of the solution set.
• The shaded part in between illustrates all the real numbers that satisfy the inequality \(-3 < x < 3\).

Graphing on a number line helps to visually confirm the range of solutions and provides a quick way to interpret the inequality.