Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$(x-3)^{2} \geq 1$$

Step-by-Step Solution

Answer

The solution to the inequality \((x-3)^2 \geq 1\) is \(x \leq 2\) or \(x \geq 4\).

Step 1: Simplify the Inequality

First, the inequality can be rewritten as \((x-3)^2 - 1 \geq 0\). This is a quadratic equation in the form of \(x^2 - bx + c\). In this case, \(b = -3\) and \(c = -1\). This equation can now be factored into \((x-3-1)(x-3+1) \geq 0\), and further simplified into \((x-4)(x-2) \geq 0\).

Step 2: Solve the Factors

Next, the factors \(x-4\) and \(x-2\) are solved separately:\n- If \(x-4 \geq 0\), then \(x \geq 4\)\n- If \(x-2 \geq 0\), then \(x \geq 2\).

Step 3: Determine the Solution Range

Lastly, analyze the signs of each interval on the number line split by the points 2 and 4, considering their inequalities:\n- If \(x < 2\), both factors will be negative, hence their product is positive.\n- If \(2 \leq x < 4\), only the factor \(x - 2\) is positive, hence the product is negative.\n- If \(x \geq 4\), both factors are positive, so their product is positive.\nThe inequality must be greater than or equal to 0, so the solution is \(x \leq 2\) or \(x \geq 4\).

Key Concepts

Quadratic Inequality

Quadratic inequalities are mathematical expressions that involve a quadratic function set to be greater than or less than a specific value. This type of inequality looks like a regular quadratic equation but with an inequality sign (>, <, ≥, or ≤). For example, \[(x-3)^2 \geq 1\] involves squaring a binomial and comparing it to 1.
Solving quadratic inequalities usually involves rearranging terms and factoring to find critical points. These critical points help determine intervals on a number line. By testing these intervals, we can determine where the inequality holds true. The final solution explains which values satisfy the inequality.
It's essential to understand the relationship between expressions and their graphical interpretations to solve these effectively.

Number Line Graph

A number line graph is a visual representation of numbers in specific intervals on a straight line. It includes labels and markers for values that help to visually interpret solutions of inequalities.
For inequalities like \((x-4)(x-2) \geq 0\), we use the number line to determine where the expression is positive or negative. Consider this inequality’s critical points (2 and 4). These points divide the number line into different intervals:

• If \(x < 2\), both parts \((x-4)\) and \((x-2)\) are negative, making their product positive.
• If \(2 \leq x < 4\), the product is negative.
• If \(x \geq 4\), both terms are positive.

By plotting on the number line, the solution becomes clear: \(x \leq 2\) and \(x \geq 4.\) Graphical methods like this help verify algebraic solutions and enhance understanding.

Factoring Quadratics

Factoring quadratics simplifies solving inequalities and equations. Often, it involves finding two binomials that multiply to form the quadratic expression. The key is to express the quadratic as \(a(x-p)(x-q)\). In our example, \((x-4)(x-2) \geq 0\), the quadratic was simplified from the inequality \((x-3)^2 - 1 \geq 0\).
Factoring involves:

• Expanding \((x-3)^2 - 1\) as a difference of squares.
• Recognizing patterns and simplifying to \((x-4)(x-2)\).

Effective factoring reveals roots or zeros, which are crucial for setting intervals and solving inequalities. These roots (like 2 and 4 in our problem) indicate changes in the sign of the expression. Understanding how to factor aids in efficiently solving quadratic inequalities.

Graphing Utilities

Graphing utilities like calculators and software assist in visually confirming solutions to quadratic inequalities. They provide a graphical depiction, which can validate whether algebraic solutions are accurate.
Let's consider the inequality \((x-4)(x-2) \geq 0\). Using graphing tools, you can create a graph of the quadratic function and see where the graph sits above the x-axis (i.e., where the product is non-negative).
These tools allow:

• Checking the solution \(x \leq 2\) and \(x \geq 4\) visually on the graph.
• Identifying points of intersection with the x-axis easily.

Ultimately, graphing utilities bridge the gap between algebraic manipulation and visual comprehension, making it easier to fully understand and verify solutions.