Problem 18 - College Algebra

Question

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ 4 x^{2}+1 \geq 4 x $$

Step-by-Step Solution

Answer

The solution is denoted by the interval notation that satisfies the inequality.

Step 1: Rearrange the inequality

Bring all terms on one side of the inequality: \(4x^2 - 4x + 1 \geq 0\). Then simplify the inequality by factoring out 4: \(x^2 - x + \frac{1}{4} \geq 0\).

Step 2: Solve the Quadratic Equation

For this step, factor the equation \(x^2 - x + \frac{1}{4} = 0\). This can be factored into \( (x- \frac{1}{2})^2 = 0 \). The solution to this equation is \(x = \frac{1}{2}\). These roots will be used to break up the number line into intervals.

Step 3: Testing the Intervals

Now break the number line up into intervals based on the roots. Here we only have one root, \(x = \frac{1}{2}\). So, we get two intervals to test: \(-\infty < x < \frac{1}{2}\) and \(\frac{1}{2}< x < \infty\). Choose any point in each interval and substitute these values into the original inequality. If the inequality is satisfied, that means the interval is part of the solution.

Step 4: Expressing the Solution in Interval Notation

Once identified the intervals which satisfy the inequality, express them in interval notation. If the interval \(-\infty < x < \frac{1}{2}\) satisfies the inequality, it is interpreted as \((-\infty, \frac{1}{2})\). If \(\frac{1}{2} < x < \infty\) satisfies the inequality, it is represented as \((\frac{1}{2}, \infty)\).

Step 5: Conclude

Combine the intervals that satisfy the inequality to get the final solution. It is important to double-check the solutions against the original inequality to make sure that the interval is valid.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions set in a form similar to quadratic equations but with inequality signs such as \( <, >, \leq, \) or \( \geq \). Unlike quadratic equations, the solutions to quadratic inequalities aren't just numbers; they are sets of numbers, often expressed in the form of intervals. The primary goal when solving these inequalities is to find out where the quadratic expression is either positive or negative, based on the inequality presented.

In our problem, we start by rearranging the inequality. We form a standard quadratic expression on one side, like in the example \(4x^2 - 4x + 1 \geq 0\). This step involves moving all terms to one side of the inequality so that our inequality is compared directly to zero. Once we have our quadratic inequality in this format, we can analyze it using methods similar to those used in solving quadratic equations.

Next, we may need to factor or simplify the expression, examine the critical points—where the expression equals zero—and determine which intervals of the variable satisfy the inequality. This helps us understand where on the number line the solution resides. The process of solving quadratic inequalities is crucial for understanding more complex algebraic problems and for graphically representing solution sets on a real number line.

Interval Notation

In mathematics, especially when dealing with inequalities, interval notation is used to express the solution set. Interval notation provides a concise way to describe a set of numbers between two endpoints.

When solving inequalities such as \(4x^2 + 1 \geq 4x\), final solutions are expressed using intervals. There are different components in interval notation:

• \((a, b)\) - denotes all numbers greater than \(a\) and less than \(b\). Neither \(a\) nor \(b\) are included in the interval.
• \([a, b] \) - includes both \(a\) and \(b\), meaning they are part of the solution set where the endpoints are part of the solution.
• \((a, b] \) or \([a, b)\) - represents intervals where one endpoint is included, and the other isn't.
• \(-\infty\) and \(\infty\) - used to describe intervals that go on indefinitely. These symbols are always paired with a parenthesis \(()\) because infinity isn't a specific number.

Using interval notation is both a convention and necessity in representing the regions of real numbers that satisfy inequalities, helping to provide a clear summary of the solution set for quadratic inequalities.

Factoring Quadratic Equations

Factoring quadratic equations is a key step in solving quadratic inequalities. To factor a quadratic equation is to express it as the product of its roots. The quadratic in this exercise, \(x^2 - x + \frac{1}{4} = 0\), was factored into \((x-\frac{1}{2})^2 = 0\).

This means our equation has what is called a repeated root at \(x = \frac{1}{2}\). Factoring helps identify these roots, or critical points, and these points are essential in determining where an inequality holds true or false. By understanding where the quadratic equals zero, we can break down the number line into different intervals and test which intervals satisfy the inequality.

Here is how the factoring process typically works:

• Identify coefficients and form: standard quadratic form \(ax^2 + bx + c\).
• Use factoring techniques, possibly converting or completing the square.
• Identify roots to construct a factored form.
• Evaluate a sample from each interval around the roots to check satisfaction of the inequality.

These steps provide clarity in breaking down polynomial behaviors and simplify the process of solving and graphing inequalities.