Problem 49 - College Algebra

Question

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$6

Step-by-Step Solution

Answer

The solution to the inequality is \(3

Step 1: Split the Compound Inequality

The compound inequality \(6

Step 2: Solve Each Inequality

For each inequality, subtract 3 on both sides to isolate x. For the first one, it yields \(x>6-3\), which simplifies to \(x>3\). The second one gives us \(x<8-3\), which simplifies to \(x<5\). Therefore, the solution of the initial inequality is \(3

Step 3: Express the Solution Set in Interval Notation and Draw it on A Number Line

The solution set in interval notation is \((3, 5)\). It includes all numbers greater than 3 and less than 5 but does not include 3 and 5 themselves. On a number line, this range is marked with open circles at the points 3 and 5, with a shaded or solid line between.

Key Concepts

Understanding Compound Inequalities

Compound inequalities are mathematical expressions that involve two separate inequalities combined into one statement using the "and" or "or" conjunctions. In our example, the compound inequality is written as \(6 < x + 3 < 8\). This tells us that \(x + 3\) must be greater than 6 while also being less than 8 simultaneously.

• "And" Compound Inequalities: These require a solution that satisfies both inequalities at the same time, just like our example.
• "Or" Compound Inequalities: These involve finding a solution that satisfies at least one of the inequalities, allowing greater flexibility.

To solve a compound inequality, you typically split it into two separate inequalities and solve them individually, ensuring that their solutions overlap. In our case, when we break down \(6 < x + 3 < 8\), it becomes \(6 < x+3\) and \(x+3 < 8\). This separation helps in simplifying the expressions for easier handling.

Mastering Interval Notation

Interval notation offers a concise way to express a range of values that satisfy an inequality. It uses brackets and parentheses to describe the starting and ending points of an interval:

• "(" or ")" denotes that the endpoint is not included in the interval, known as open interval.
• "[" or "]" indicates that the endpoint is included in the interval, known as closed interval.

For example, the solution \(3 < x < 5\) is written in interval notation as \((3, 5)\). This notation clearly shows that 3 and 5 are not part of the solution set. Using interval notation provides a straightforward way to convey the entire set of solutions without listing each value individually. It's a widely accepted method in mathematics for denoting intervals.

Graphing on a Number Line

A number line graph helps to visually represent the solution of an inequality. This visualization makes it easier to understand which numbers are part of the solution set.To draw a number line for the solution \((3, 5)\):

• Begin by marking the numbers 3 and 5 on the line.
• Use open circles to indicate that these endpoints are not included in the solution.
• Draw a solid line or shade the region between the open circles to represent all the numbers between 3 and 5.

This clear depiction on the number line confirms the solution \((3, 5)\) by showing the continuous set of values that satisfy the inequality \(3 < x < 5\). Number line graphs are intuitive tools that complement algebraic and interval notation solutions.

Isolating Variables for Solution

Isolating the variable is a crucial step in solving inequalities. It involves rearranging the inequality such that the variable (in this case, \(x\)) stands alone on one side, making it easier to identify its possible values.For \(6 < x+3 < 8\), we split it into two inequalities:

• From \(6 < x + 3\), subtract 3 from both sides to get \(x > 3\).
• From \(x + 3 < 8\), similarly subtract 3 from both sides to find \(x < 5\).

These steps allow us to effectively "unlock" the variable, leaving us with the clearer and manageable statements \(x > 3\) and \(x < 5\). Isolating variables is an essential skill, whether working with equations or inequalities, ensuring the problem is simplified correctly and efficiently.

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Other exercises in this chapter

Problem 49

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Problem 49

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Problem 49

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Key Concepts

Understanding Compound Inequalities

Compound inequalities are mathematical expressions that involve two separate inequalities combined into one statement using the "and" or "or" conjunctions. In our example, the compound inequality is written as \(6 < x + 3 < 8\). This tells us that \(x + 3\) must be greater than 6 while also being less than 8 simultaneously.

• "And" Compound Inequalities: These require a solution that satisfies both inequalities at the same time, just like our example.
• "Or" Compound Inequalities: These involve finding a solution that satisfies at least one of the inequalities, allowing greater flexibility.

To solve a compound inequality, you typically split it into two separate inequalities and solve them individually, ensuring that their solutions overlap. In our case, when we break down \(6 < x + 3 < 8\), it becomes \(6 < x+3\) and \(x+3 < 8\). This separation helps in simplifying the expressions for easier handling.

Mastering Interval Notation

Interval notation offers a concise way to express a range of values that satisfy an inequality. It uses brackets and parentheses to describe the starting and ending points of an interval:

• "(" or ")" denotes that the endpoint is not included in the interval, known as open interval.
• "[" or "]" indicates that the endpoint is included in the interval, known as closed interval.

For example, the solution \(3 < x < 5\) is written in interval notation as \((3, 5)\). This notation clearly shows that 3 and 5 are not part of the solution set. Using interval notation provides a straightforward way to convey the entire set of solutions without listing each value individually. It's a widely accepted method in mathematics for denoting intervals.

Graphing on a Number Line

A number line graph helps to visually represent the solution of an inequality. This visualization makes it easier to understand which numbers are part of the solution set.To draw a number line for the solution \((3, 5)\):

• Begin by marking the numbers 3 and 5 on the line.
• Use open circles to indicate that these endpoints are not included in the solution.
• Draw a solid line or shade the region between the open circles to represent all the numbers between 3 and 5.

This clear depiction on the number line confirms the solution \((3, 5)\) by showing the continuous set of values that satisfy the inequality \(3 < x < 5\). Number line graphs are intuitive tools that complement algebraic and interval notation solutions.

Isolating Variables for Solution

Isolating the variable is a crucial step in solving inequalities. It involves rearranging the inequality such that the variable (in this case, \(x\)) stands alone on one side, making it easier to identify its possible values.For \(6 < x+3 < 8\), we split it into two inequalities:

• From \(6 < x + 3\), subtract 3 from both sides to get \(x > 3\).
• From \(x + 3 < 8\), similarly subtract 3 from both sides to find \(x < 5\).

These steps allow us to effectively "unlock" the variable, leaving us with the clearer and manageable statements \(x > 3\) and \(x < 5\). Isolating variables is an essential skill, whether working with equations or inequalities, ensuring the problem is simplified correctly and efficiently.