Problem 28 - 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. $$ 4 x^{2}-4 x+1 \geq 0 $$

Step-by-Step Solution

Answer

The solution set of the inequality \(4x^2 - 4x + 1 \geq 0\) is \((- \infty, 0.5] \cup [0.5, \infty)\).

Step 1: Understand the Inequality

The inequality to solve is \(4x^2 - 4x + 1 \geq 0\). It is already in the standard form \(ax^2 + bx + c\).

Step 2: Find The Discriminant

The discriminant, which is \(b^2 - 4ac\), helps to identify the number and nature of the roots of a quadratic. Here, \(a = 4\), \(b = -4\), and \(c = 1\). Therefore, the discriminant is \((-4)^2 - 4*4*1 = 16 - 16 = 0\).

Step 3: Find The Roots

Since the discriminant is equal to zero, this indicates that there is exactly one real root for the quadratic equation \(4x^2 - 4x + 1 = 0\), which we can be calculated using the quadratic formula, \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This root will serve as our critical point. Solving for \(x\), we have \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4*4*1}}{2*4} = \frac{4}{8} = 0.5\)

Step 4: Test the Intervals

We have one critical point, \(x = 0.5\), which divides the number line into two intervals. To determine the solution set of the inequality, select any number from each interval and substitute into the original inequality \(4x^2 - 4x + 1 \geq 0\). If the inequality holds true, then the interval is part of the solution set. Choosing numbers \(0\) and \(1\) for the intervals \((- \infty, 0.5)\) and \((0.5, \infty)\), after substitution and simplification both values hold true. Therefore, both intervals are in the solution set.

Step 5: Write the Solution in Interval Notation and Draw a Number Line

The solution to the inequality \(4x^2 - 4x + 1 \geq 0\) in interval notation is \((- \infty, 0.5] \cup [0.5, \infty)\). A number line will contain a closed dot on \(0.5\) (inclusive \(\geq\)) and arrows pointing towards \(-\infty\) and \(+\infty\).

Key Concepts

Discriminant

The discriminant is a crucial component when dealing with quadratic equations, particularly in solving polynomial inequalities. It is given by the formula \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the standard form of a quadratic equation \(ax^2 + bx + c\). The discriminant can tell you:

• If it is positive: there are two distinct real roots.
• If it is zero: there is exactly one real root, or a double root.
• If it is negative: there are no real roots, only complex ones.

In this exercise, the discriminant is zero, which indicates a unique real solution (or a double root). This special case means that the quadratic touches the x-axis at just one point. This point is not only a root but is also significant for testing intervals in inequalities.

Quadratic Formula

The quadratic formula is an essential tool for finding the roots of any quadratic equation in the form \(ax^2 + bx + c = 0\). It is derived from completing the square and is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to calculate the solutions of the equation, whether they are real or complex. The \(\pm\) sign indicates that there can be up to two roots - one by adding and one by subtracting the square root part. In our case, using \(a = 4\), \(b = -4\), and \(c = 1\), the formula yields just one root, \(x = 0.5\), since the discriminant is zero. This particular root is essential for plotting and solving the inequality, as it serves as the critical point.

Interval Notation

Interval notation is a concise way of writing subsets of the real number line. For inequalities, like the ones involving quadratics, we use it to represent solution sets comprehensively. In this form, brackets and parentheses are used:

• Parentheses \(( )\) indicate that the endpoint is not included, representing "open intervals."
• Brackets \([ ]\) signify that the endpoint is included, representing "closed intervals."

For the inequality \(4x^2 - 4x + 1 \geq 0\), the solution is expressed in interval notation as \((-\infty, 0.5] \cup [0.5, \infty)\). Here, \(0.5\) is included due to the \(\geq\) sign, indicating equality holds, with the union \(\cup\) combining both intervals as part of the solution.

Critical Point

A critical point in the context of polynomial inequalities is a value that separates different behavior of the function. For quadratic functions, roots often serve as critical points. They are the values where the inequality can change from true to false, or vice versa.In our exercise, the critical point is \(x = 0.5\). This particular value is where the quadratic function \(4x^2 - 4x + 1\) equals zero. Testing intervals>

• Choose numbers on either side of the critical point to see if they satisfy the inequality.
• For \((-\infty, 0.5)\), try a number like \(x = 0\).
• For \((0.5, \infty)\), try \(x = 1\).

Both test points should satisfy the original inequality, affirming both intervals are part of the solution with a closed dot, indicating inclusion of \(x = 0.5\) in the solution set.