Problem 63 - College Algebra

Question

Solve absolute value inequality. \(|2 x-6|<8\)

Step-by-Step Solution

Answer

The solution to the inequality \(|2x-6|<8\) is \(-1

Step 1: Understand Absolute Value Inequality

An absolute value is always positive or zero, never negative. In the inequality \(|2x-6|<8\), we want to find the values of x such that the absolute value of \(2x-6\) is less than 8.

Step 2: Break into Two Inequalities

We break the absolute value inequality into two separate inequalities: \(2x-6<8\) and \(2x-6>-8\). This is based on the definition of absolute value. An absolute value of a number is the number itself or its opposite.

Step 3: Solve First Inequality

Solve the inequality \(2x-6<8\). Add 6 to both sides to isolate \(2x\), resulting to \(2x<14\). Divide 2 on both sides to solve for \(x\), which gives \(x<7\).

Step 4: Solve Second Inequality

Solve the inequality \(2x-6>-8\). Add 6 to both sides to isolate \(2x\), which gives \(2x>-2\). Divide 2 on both sides to solve for \(x\), this will result in \(x>-1\).

Step 5: Combine Solutions

The solution for \(x\) must satisfy both \(x<7\) and \(x>-1\). The values that satisfy both these inequalities are between -1 and 7. So, the solution is \(-1

Key Concepts

Inequalities

Inequalities are expressions that involve a comparison between two values or expressions. They tell us how one quantity relates to another. For example, the inequality \( x < 7 \) means that \( x \) can be any number less than 7.

This concept is different from an equation, as it does not imply equality but rather a range of possibilities. Inequalities use symbols such as:

• '>' meaning greater than
• '<' meaning less than
• '\geq' meaning greater than or equal to
• '\leq' meaning less than or equal to

Solving inequalities often results in a range of values instead of a single solution, allowing us to understand a broader set of possibilities.

Mastering inequalities is essential for solving real-world problems in areas like engineering and economics, where limits and ranges are frequent.

Solving Inequalities

Solving inequalities involves finding the values that make the inequality true. It's similar to solving equations, but with some differences. Let’s see how it’s done:

1. **Perform similar operations** as you would when solving equations, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number.
2. **Important Rule**: If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example, \(-x > -3\) becomes \(x < 3\) when both sides are divided by -1.
3. **Combining Solutions**: When dealing with compound inequalities, you may need to find the set of values that satisfy all parts of the inequality system.

In our exercise, once we solve each part of the absolute value inequality, we combine them to find that \(-1 < x < 7\). This tells us x is greater than -1 and less than 7.

Absolute Value Concepts

The absolute value of a number refers to its distance from zero on the number line, without considering direction. This value is always non-negative. In notation, the absolute value of \(x\) is written as \(|x|\).

The key property of absolute values is:

• \( |a| = a \) if \( a \geq 0 \)
• \( |a| = -a \) if \( a < 0 \)

When dealing with absolute value inequalities like \(|2x-6|<8\), we translate them into two separate inequalities:

1. The expression inside the absolute value is less than the given number (\(2x-6<8\)).
2. The expression inside the absolute value is greater than the negative of that number (\(2x-6>-8\)).

This ensures that we cover both possible cases for the expressions that fit within the absolute value.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operations to represent a quantity. They are fundamental in solving equations and inequalities.

When working with expressions like \(2x - 6\), we perform operations to isolate the variable. For example:

• Addition or subtraction to clear constants from one side and leave the variable on the other.
• Multiplication or division to solve for the variable itself.

In our original exercise, we manipulate the expression \(2x - 6\) by first adding 6 to isolate the term with the variable, then dividing by 2 to solve for x.

Understanding how to manipulate algebraic expressions is key in simplifying complex equations and inequalities, making them easier to solve. This simplifies the process and strengthens problem-solving skills in mathematics.

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Problem 63

Other exercises in this chapter

Problem 62

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

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Problem 62

Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.

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Problem 63

Solve each equation in Exercises \(47-64\) by completing the square. $$3 x^{2}-2 x-2=0$$

View solution

Problem 63

Explain why \((5,-2)\) and \((-2,5)\) do not represent the same point. CAN'T COPY THE GRAPH

View solution

Key Concepts

Inequalities

Inequalities are expressions that involve a comparison between two values or expressions. They tell us how one quantity relates to another. For example, the inequality \( x < 7 \) means that \( x \) can be any number less than 7.

This concept is different from an equation, as it does not imply equality but rather a range of possibilities. Inequalities use symbols such as:

• '>' meaning greater than
• '<' meaning less than
• '\geq' meaning greater than or equal to
• '\leq' meaning less than or equal to

Solving inequalities often results in a range of values instead of a single solution, allowing us to understand a broader set of possibilities.

Mastering inequalities is essential for solving real-world problems in areas like engineering and economics, where limits and ranges are frequent.

Solving Inequalities

Solving inequalities involves finding the values that make the inequality true. It's similar to solving equations, but with some differences. Let’s see how it’s done:

1. **Perform similar operations** as you would when solving equations, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number.
2. **Important Rule**: If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example, \(-x > -3\) becomes \(x < 3\) when both sides are divided by -1.
3. **Combining Solutions**: When dealing with compound inequalities, you may need to find the set of values that satisfy all parts of the inequality system.

In our exercise, once we solve each part of the absolute value inequality, we combine them to find that \(-1 < x < 7\). This tells us x is greater than -1 and less than 7.

Absolute Value Concepts

The absolute value of a number refers to its distance from zero on the number line, without considering direction. This value is always non-negative. In notation, the absolute value of \(x\) is written as \(|x|\).

The key property of absolute values is:

• \( |a| = a \) if \( a \geq 0 \)
• \( |a| = -a \) if \( a < 0 \)

When dealing with absolute value inequalities like \(|2x-6|<8\), we translate them into two separate inequalities:

1. The expression inside the absolute value is less than the given number (\(2x-6<8\)).
2. The expression inside the absolute value is greater than the negative of that number (\(2x-6>-8\)).

This ensures that we cover both possible cases for the expressions that fit within the absolute value.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operations to represent a quantity. They are fundamental in solving equations and inequalities.

When working with expressions like \(2x - 6\), we perform operations to isolate the variable. For example:

• Addition or subtraction to clear constants from one side and leave the variable on the other.
• Multiplication or division to solve for the variable itself.

In our original exercise, we manipulate the expression \(2x - 6\) by first adding 6 to isolate the term with the variable, then dividing by 2 to solve for x.

Understanding how to manipulate algebraic expressions is key in simplifying complex equations and inequalities, making them easier to solve. This simplifies the process and strengthens problem-solving skills in mathematics.