Problem 23 - Precalculus : Building Concepts and Connections

Question

Solve the inequality by factoring. $$5 x^{2}-8 x \geq 4$$

Step-by-Step Solution

Answer

The solutions to the inequality are \( x \leq 0.8 \) or \( x \geq 1 \)

Step 1: Rearrange the inequality

Start by rearranging the inequality and setting it to zero. Subtract 4 from both sides to get: \(5 x^{2} - 8 x - 4 \geq 0\)

Step 2: Factor the quadratic

Now, factor the quadratic. You can factor it as follows: \(5 x^{2} - 8 x - 4 = (5x - 4)(x - 1) \)

Step 3: Find the critical points

Next, find the critical points of the inequality, which occur when the quadratic equals to zero. We get the two critical points by setting each factor equal to zero: \(5x - 4 = 0\) and \(x - 1 = 0\), which gives \(x = 0.8\) and \(x = 1\)

Step 4: Test the intervals

Now test the intervals determined by the critical points. Choosing test points from each interval to put into the original inequality will help decide which intervals are part of the solution. Try \(x = 0.5\), \(x = 0.9\), and \(x = 2\). You will find that \(x = 0.5\), and \(x = 2\) are each greater than 0, while \(x = 0.9\) is not. Thus the solution to the inequality is \( x \leq 0.8 \) or \( x \geq 1 \).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to simplify a quadratic expression by expressing it as a product of two simpler expressions. This is a powerful tool for solving quadratic equations or inequalities. When the quadratic is in the form of \( ax^2 + bx + c \), the goal is to express it as \((px + q)(rx + s)\), where \(p, q, r,\) and \(s\) are numbers that satisfy the original equation. To factor a quadratic like \(5x^2 - 8x - 4\), you need to find two numbers that multiply to give the product \(a \times c = -20\) and add up to \(b = -8\). In this particular example, the expression is successfully factored into \((5x - 4)(x - 1)\).

• Start by identifying the coefficients \(a\), \(b\), and \(c\) of the quadratic.
• Find a pair of numbers that multiply to \(a \times c\) and sum to \(b\).
• Rewrite the middle term using these two numbers and regroup the terms if needed before factoring by grouping to achieve the final factorization.

Factoring can be seen as "undoing" multiplication, and it makes other steps, like finding critical points, much simpler.

Critical Points

Critical points in solving quadratic inequalities are values of \(x\) where the quadratic expression equals zero. They are crucial because they split the number line into intervals where the expression either satisfies or does not satisfy the inequality. Once a quadratic is factored, like \((5x - 4)(x - 1)\), the critical points are determined by setting each factor equal to zero.

• For \(5x - 4 = 0\), we find \(x = 0.8\).
• For \(x - 1 = 0\), we find \(x = 1\).

These points are essential as they mark the boundaries of intervals that need to be tested for the inequality. Critical points serve as the transition between where the quadratic inequality flips from true to false or vice versa. Recognizing these points allows us to break the problem into manageable parts.

Interval Testing

Interval testing is a step where you determine which regions (defined by the critical points) satisfy the inequality. This involves picking test values from intervals split by these critical points. Each interval represents a different segment on the number line where the inequality might behave differently. For this quadratic \((5x - 4)(x - 1) \geq 0\), the critical points divide the line into intervals: \((-\infty, 0.8)\), \((0.8, 1)\), and \((1, \infty)\).

• Choose a test point from each region, like \(x = 0.5\), \(x = 0.9\), and \(x = 2\).
• Plug these numbers into the factored inequality \((5x - 4)(x - 1)\).
• Determine if the result is positive or negative to see if the inequality holds.

Testing intervals is like simple detective work; it reveals where the inequality is true. Once completed, compile the results to identify the solution set: in this case, \(x \leq 0.8\) or \(x \geq 1\). This process ensures accuracy in finding the solution to the inequality.