Problem 51 - College Algebra with Modeling and Visualization

Question

Solve the inequality. $$ 7 x^{2}+515.2 \geq 179.8 x $$

Step-by-Step Solution

Answer

The solution is \( x \leq 4.68 \) or \( x \geq 15.60 \).

Step 1: Rearrange the Inequality

Start by rearranging the given inequality into the standard quadratic form. The original inequality is: \[ 7x^2 + 515.2 \geq 179.8x \]Subtract \( 179.8x \) from both sides to get:\[ 7x^2 - 179.8x + 515.2 \geq 0 \]

Step 2: Identify Coefficients for Quadratic Formula

The inequality \( 7x^2 - 179.8x + 515.2 \geq 0 \) is in the form \( ax^2 + bx + c \geq 0 \). Here, the coefficients are:- \( a = 7 \)- \( b = -179.8 \)- \( c = 515.2 \)

Step 3: Calculate the Discriminant

To determine the roots of the equation, calculate the discriminant, \( D \), using the formula:\[ D = b^2 - 4ac \]Substitute the values:\[ D = (-179.8)^2 - 4 \times 7 \times 515.2 \]\[ D = 32392.04 - 14425.6 \]\[ D = 17966.44 \]

Step 4: Solve for the Roots Using the Quadratic Formula

Since the discriminant is positive, there are two distinct real roots. Use the quadratic formula:\[ x = \frac{-b \pm \sqrt{D}}{2a} \]Calculate the roots:\[ x_1 = \frac{179.8 + \sqrt{17966.44}}{14} \] \[ x_2 = \frac{179.8 - \sqrt{17966.44}}{14} \] Compute these values to find:\[ x_1 \approx 15.60 \]\[ x_2 \approx 4.68 \]

Step 5: Determine the Solution Set for the Inequality

The quadratic inequality \( 7x^2 - 179.8x + 515.2 \geq 0 \) is satisfied outside the interval defined by the roots due to its leading coefficient (\( a = 7 > 0 \)). Therefore, the solution set for \( x \) is:\[ x \leq 4.68 \text{ or } x \geq 15.60 \]

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations, expressed as \( ax^2 + bx + c = 0 \). When rearranging an inequality into this quadratic form by moving all terms to one side, you can utilize the formula

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This allows you to find the roots, or solutions, of the quadratic equation, which also serve as important points when solving quadratic inequalities. For our exercise, the coefficients are \( a = 7 \), \( b = -179.8 \), and \( c = 515.2 \). By plugging these values into the quadratic formula, you can find the values that make the quadratic expression equal to zero. These roots help identify the range of solutions for the inequality. Always remember the plus/minus symbol indicates you will get two solutions, known as the roots of the quadratic equation.

Discriminant

The discriminant is a critical part of the quadratic formula expressed as \( D = b^2 - 4ac \). It's a powerful indicator of the nature of the roots of a quadratic equation without actually finding them.

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is exactly one real root (a repeated root).
• If \( D < 0 \), there are no real roots, only complex roots.

For our example, we calculated the discriminant to be 17966.44, which is greater than zero. This tells us the quadratic equation has two distinct real roots. Knowing the nature of the roots helps you determine where the inequality holds, as it divides the number line into intervals that can be tested.

Inequality Solutions

Solving quadratic inequalities involves finding the values of \( x \) that satisfy the condition given by the inequality. After finding the roots using the quadratic formula, you need to determine where the quadratic expression is greater than or less than zero based on the inequality symbols. For the inequality \( 7x^2 - 179.8x + 515.2 \geq 0 \), we found the roots to be approximately \( x_1 \approx 4.68 \) and \( x_2 \approx 15.60 \).

• Since the leading coefficient \( a = 7 \) is positive, the parabola opens upwards, meaning it is above the x-axis outside the interval defined by the roots.
• Therefore, the values of \( x \) that satisfy the inequality are \( x \leq 4.68 \) or \( x \geq 15.60 \).

The solution involves checking the intervals created by the roots and determining which intervals satisfy the inequality condition. This leads to understanding whether the expression is positive or negative in various regions of the 'x' axis.