Problem 58 - Intermediate Algebra

Question

Explain how to solve the inequality \((x-2)^{2}>0\) by inspection.

Step-by-Step Solution

Answer

The inequality holds for all \(x\neq 2\).

Step 1: Identify the Structure of the Inequality

Look at the inequality \((x-2)^{2}>0\). Notice that it's a square of a binomial. A square of any real number is always non-negative (positive or zero). Here, we need to find when it is strictly positive.

Step 2: Determine When the Expression Equals Zero

The expression \((x-2)^{2}\) equals zero when \(x-2=0\). Solve this to find \(x=2\). This is the only point where the square can be zero.

Step 3: Establish When the Expression is Positive

Since \((x-2)^{2}\) is zero only at \(x=2\), it is positive for all \(x\) different from 2. Hence, the expression \((x-2)^{2} > 0\) for \(x<2\) and \(x>2\).

Step 4: Write the Solution Set

The solution set consists of all real numbers except \(x=2\). In interval notation, this is expressed as \(x \in (-\infty, 2) \cup (2, \infty)\).

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression and establishes a range where it holds true. For instance, in the inequality

• \((x-2)^2 > 0\),

we don't want the expression to simply "equal" something. Instead, we are more interested in identifying the set of values for which the expression is greater than zero.
To solve this by inspection:

• Recognize that \((x-2)^2\) represents a perfect square.
• A perfect square is always non-negative, meaning it's anything but negative, but can be zero.
• Hence, the inequality \((x-2)^2 > 0\) prompts us to find when \( (x-2)^2 \) is positive. It's positive when \(x eq 2\), which is everywhere except at \(x = 2\).

Quadratic inequalities often require you to think about where the graph of the quadratic touches or crosses the x-axis, and this forms the basis for understanding the solution set.

Solution Set

The solution set refers to all the values of \(x\) that satisfy a given inequality. For quadratic inequalities such as \((x-2)^2 > 0\), this means identifying all points where the expression is greater than zero.
To clarify this:

• Determine where \((x-2)^2 = 0\). This happens only when \(x = 2\).
• Since the inequality asks for when the expression is strictly greater than zero, the solution set includes every other real number apart from \(x = 2\).

The solution set can sometimes be shown graphically or through number line diagrams to make visualizing easier. It generally consists of intervals or a union of intervals where the inequality is satisfied.

Interval Notation

When we have set out to express the solution set of an inequality clearly, we commonly use interval notation. Let’s consider the solution set we found for the inequality \((x-2)^2>0\):
The key elements of interval notation include:

• Round parentheses \((\text{ or } )\) indicate that the endpoint is not included in the interval.
• Infinity symbols \((-\infty\text{ or } \infty)\) indicate the set extends without bound in either positive or negative direction.

Therefore, the solution set \(x \in (-\infty, 2) \cup (2, \infty)\) suggests all real numbers except \(x=2\). This concise notation effectively communicates that we are dealing with two separate intervals. Here,

• \((-\infty, 2)\) denotes all numbers\(x\) less than 2,
• while \((2, \infty)\) denotes all numbers\(x\) greater than 2.

This makes it straightforward for anyone reviewing the solution to quickly grasp the full range of solutions.