Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to solve the inequality. (Round each number in your answer to two decimal places.) \(1.2 x^{2}+4.8 x+3.1<5.3\)

Step-by-Step Solution

Answer

The solution to the inequality are the set of values for \(x\) found in Step 3, rounded to two decimal places.

Step 1: Rearrange the inequality

First we rearrange the inequality in the form \(ax^2+bx+c<0\) by subtracting \(5.3\) from both sides. This results in the equation \(1.2x^2 + 4.8x + 3.1 - 5.3<0\), which simplifies to \(1.2x^2 + 4.8x - 2.2<0\)

Step 2: Determine the roots

We then determine the roots of the equation \(1.2x^2 + 4.8x - 2.2=0\) using the quadratic formula \(x = [-b \pm \sqrt{(b^2 - 4ac)}]{/2a}\). Substituting in the values \(a=1.2\), \(b=4.8\), and \(c=-2.2\) gives \(x = [-4.8 \pm \sqrt{(4.8^2 - 4*1.2*(-2.2))}]{/2*1.2}\), which simplifies to \(x = [-4 \pm \sqrt{23.04+10.56}]{/2.4}\) before finally calculating the roots.

Step 3: Construct the number line and find the range for x

We plot the roots obtained in Step 2 on the number line, along with testing the intervals for values of \(x\) which satisfy \(1.2x^2 + 4.8x - 2.2<0\).

Key Concepts

Quadratic Formula

The quadratic formula is a critical tool in solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It is particularly useful when the equation doesn't factor easily. The formula provides the roots of the quadratic equation and is expressed as:

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

In this formula:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

To apply the quadratic formula, you substitute the values of \( a \), \( b \), and \( c \) from the quadratic equation and solve for \( x \). The discriminant, \( b^2 - 4ac \), under the square root sign, determines the nature of the roots. If it's positive, you'll get two distinct real roots. If it's zero, you have one real root. A negative discriminant means the roots are complex and not real numbers.

Roots of Equations

The roots of an equation are the values of \( x \) for which the equation equals zero. In the context of quadratic equations, finding roots is synonymous with finding the point(s) where the parabola intersects the x-axis.For the equation \( ax^2 + bx + c = 0 \), the roots can be determined using either factorization, completing the square, or the quadratic formula. When using the quadratic formula, the solution \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) directly gives you the roots.Understanding the nature and position of these roots is essential:

• If the equation has two distinct roots, the parabola will intersect the x-axis at two points.
• If it has one root, or a repeated root, the parabola will just touch the x-axis.
• If the equation has no real roots, the parabola does not intersect the x-axis.

Determining where the curve lies relative to the x-axis helps you understand the behavior of quadratic inequalities.

Polynomial Inequalities

Polynomial inequalities involve expressions with polynomials being compared, usually using signs like \( < \), \( > \), \( \leq \), or \( \geq \). Solving these inequalities involves several steps to ensure accurate solutions.For example, let's solve the inequality \( 1.2x^2 + 4.8x - 2.2 < 0 \). First, you would find the roots of the corresponding equation \( 1.2x^2 + 4.8x - 2.2 = 0 \) using the quadratic formula. After determining the roots, the solution involves analyzing intervals on a number line:

• Plot the roots on a number line.
• Test intervals between and outside the roots to determine where the inequality holds true.
• Based on these tests, conclude where the polynomial is less than zero.

It's important to note that the intervals where the polynomial is negative or positive heavily depend on the leading coefficient \( a \) and the nature of the roots. This visualization helps in determining the solution to the inequality.