Problem 96 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. Solve: \((x-3)^{2}>0\)

Step-by-Step Solution

Answer

The solution to the inequality \((x-3)^{2} > 0\) is \(x<3\) or \(x>3\). In interval notation, it is \(-\infty

Step 1: Simplify the Expression

The expression is already in its simplest form, so no simplification is required. \((x - 3)^{2}>0\) implies that \(x - 3\) is either greater than 0 or less than 0.

Step 2: Solve Inequalities Seperately

Solve the inequality \(x - 3 > 0\) and \(x - 3 < 0\) separately. Solving the first inequality we find \(x > 3\). Solving the second inequality we find \(x < 3\).

Step 3: Combine the Results

Notice that we cannot have \(x=3\) since \(0>0\) is not a valid inequality. This indicates the value \(x=3\) is not included in the solution. Hence the solution to this inequality is \(x<3\) or \(x>3\). For this reason, we usually write the solution in interval notation as: \(-\infty

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared to zero or another expression using inequality symbols like ">" or "<". Unlike quadratic equations, which set the expression equal to zero, quadratic inequalities determine a range of values for the variable that satisfy the condition.

To solve a quadratic inequality, follow these steps:

• Simplify the quadratic expression if possible.
• Determine where the expression equals zero to find critical values, typically by setting the quadratic expression equal to zero and solving for the variable.
• Test intervals determined by the critical values on a number line to see if they satisfy the inequality.
• Combine the intervals where the inequality holds true.

In the context of the exercise, \((x-3)^{2}>0\), the value \(x=3\) is not included because at \(x=3\), the inequality equals zero, which is not greater than zero.

Interval Notation

Interval notation is a method used to describe a set of numbers, usually solutions to inequalities, by specifying the start and end points of intervals. This notation is more concise than using inequality symbols.

The key components of interval notation are:

• Round brackets "( )" are used if the endpoints are not included in the interval, known as open intervals.
• Square brackets "[ ]" are used if the endpoints are included, known as closed intervals.
• Infinity symbols "∞" and "-∞" are always used with round brackets because infinity itself is not a number that can be included.

For example, the solution to \((x-3)^{2}>0\) is expressed as \((-∞, 3) \cup (3, ∞)\). This indicates values less than 3 or greater than 3, excluding 3 itself.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operations (like addition, subtraction, multiplication, and division). They are fundamental components in algebra used to represent mathematical relationships or conditions.

For quadratic expressions like \((x-3)^2\), it's important to understand the role each part plays:

• The quadratic term \((x-3)^2\) suggests there's a squared variable, which is the focus of the expression.
• This specific form \((x-3)^2\) indicates a transformation of the variable \(x\), shifting the graph horizontally by 3 units.
• When reading inequalities such as \((x-3)^2 > 0\), the expression states that the distance between \(x\) and 3 is always positive except when \(x=3\), supporting the solution found earlier.

Working with algebraic expressions requires applying operations correctly and simplifying where possible to make problem-solving manageable.

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Problem 96

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Problem 96

Other exercises in this chapter

Problem 95

Solve each equation. $$2|\ln x|-6=0$$

View solution

Problem 96

Evaluate or simplify each expression without using a calculator. $$\ln e^{13 x}$$

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Problem 96

Solve each equation. $$3|\log x|-6=0$$

View solution

Problem 97

Evaluate or simplify each expression without using a calculator. $$e^{\ln 5 x^{2}}$$

View solution

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared to zero or another expression using inequality symbols like ">" or "<". Unlike quadratic equations, which set the expression equal to zero, quadratic inequalities determine a range of values for the variable that satisfy the condition.

To solve a quadratic inequality, follow these steps:

• Simplify the quadratic expression if possible.
• Determine where the expression equals zero to find critical values, typically by setting the quadratic expression equal to zero and solving for the variable.
• Test intervals determined by the critical values on a number line to see if they satisfy the inequality.
• Combine the intervals where the inequality holds true.

In the context of the exercise, \((x-3)^{2}>0\), the value \(x=3\) is not included because at \(x=3\), the inequality equals zero, which is not greater than zero.

Interval Notation

Interval notation is a method used to describe a set of numbers, usually solutions to inequalities, by specifying the start and end points of intervals. This notation is more concise than using inequality symbols.

The key components of interval notation are:

• Round brackets "( )" are used if the endpoints are not included in the interval, known as open intervals.
• Square brackets "[ ]" are used if the endpoints are included, known as closed intervals.
• Infinity symbols "∞" and "-∞" are always used with round brackets because infinity itself is not a number that can be included.

For example, the solution to \((x-3)^{2}>0\) is expressed as \((-∞, 3) \cup (3, ∞)\). This indicates values less than 3 or greater than 3, excluding 3 itself.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operations (like addition, subtraction, multiplication, and division). They are fundamental components in algebra used to represent mathematical relationships or conditions.

For quadratic expressions like \((x-3)^2\), it's important to understand the role each part plays:

• The quadratic term \((x-3)^2\) suggests there's a squared variable, which is the focus of the expression.
• This specific form \((x-3)^2\) indicates a transformation of the variable \(x\), shifting the graph horizontally by 3 units.
• When reading inequalities such as \((x-3)^2 > 0\), the expression states that the distance between \(x\) and 3 is always positive except when \(x=3\), supporting the solution found earlier.

Working with algebraic expressions requires applying operations correctly and simplifying where possible to make problem-solving manageable.