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Magazine Chapter 8: Problem 38
Solve each inequality. Write the solution set in interval notation. $$16 x^{4}-40 x^{2}+9 \leq 0$$
Short Answer
Expert verified
The solution set is \([-\frac{3}{2}, -\frac{1}{2}] \cup [\frac{1}{2}, \frac{3}{2}] \).
Step by step solution
01
Factor the Expression
First, we recognize that the inequality is a polynomial in terms of the square of a binomial. Let us rewrite the variable: make a substitution where \( u = x^2 \). Therefore, the expression becomes \( 16u^2 - 40u + 9 \). We factor this quadratic expression to find the roots. Using the quadratic formula, \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), solve for \( u \) where \( a = 16 \), \( b = -40 \), and \( c = 9 \).
02
Solve for Roots Using the Quadratic Formula
Applying the quadratic formula: \( b^2 - 4ac = (-40)^2 - 4 \times 16 \times 9 = 1600 - 576 = 1024 \). The square root of 1024 is 32. So, the roots are \( u = \frac{40 \pm 32}{32} \). This gives us roots \( u = \frac{72}{32} = \frac{9}{4} \) and \( u = \frac{8}{32} = \frac{1}{4} \).
03
Determine the Intervals
Since \( u = x^2 \), substitute back to find the intervals for \( x \). Therefore, we have \( x^2 = \frac{9}{4} \) or \( x^2 = \frac{1}{4} \). This corresponds to the values \( x = \pm\frac{3}{2} \) and \( x = \pm\frac{1}{2} \). We now calculate the intervals of \( x \) by evaluating the expression: \( 16x^4 - 40x^2 + 9 \) is less than or equal to zero for \( x \in [-\frac{3}{2}, -\frac{1}{2}] \cup [\frac{1}{2}, \frac{3}{2}] \).
04
Test the Intervals
Choose test points from each interval to determine where the inequality holds. For instance, test points in the intervals \((-\infty, -\frac{3}{2})\), \((-\frac{3}{2}, -\frac{1}{2})\), \((-\frac{1}{2}, \frac{1}{2})\), \((\frac{1}{2}, \frac{3}{2})\), and \((\frac{3}{2}, \infty)\) will show that the inequality holds in \((-\frac{3}{2}, -\frac{1}{2})\) and \((\frac{1}{2}, \frac{3}{2})\).
05
Write Solution in Interval Notation
Combine the intervals where the inequality is satisfied. Thus, the solution set is \([-\frac{3}{2}, -\frac{1}{2}] \cup [\frac{1}{2}, \frac{3}{2}] \) in interval notation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a mathematical way to describe the set of solutions that satisfy an inequality by using intervals of numbers.
It's like a special shorthand to show where a variable like \( x \) can live on the real number line.
Here's how to understand interval notation:
It's like a special shorthand to show where a variable like \( x \) can live on the real number line.
Here's how to understand interval notation:
- Square brackets \([ \text{ and } ]\) mean the endpoint is included, showing that the number is part of the solution set.
- Round brackets \(( \text{ and } )\) mean the endpoint is not included, showing the number is a boundary point, but not part of the solution.
- The union symbol \( \cup \) indicates that multiple intervals are joined together to form a solution set.
- Examples include \([1, 3] \) for numbers between 1 and 3, including both, and \((2, 4] \cup [5, 6)\) for numbers between 2 and 4, where 2 is not included but 4 is, plus numbers between 5 and 6, where 6 is not included.
Quadratic Formula
The quadratic formula is a powerful mathematical tool used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \).
Roots are the values of \( x \) that make the equation zero.
The formula is defined as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how it works:
Roots are the values of \( x \) that make the equation zero.
The formula is defined as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how it works:
- \( a \), \( b \), and \( c \) are coefficients from the quadratic equation.
- The term \( b^2 - 4ac \) is called the "discriminant." It's crucial because:
- If positive, the equation has two distinct real roots.
- If zero, the roots are real and equal or repeated.
- If negative, the roots are complex and not real.
- The "\( \pm \)" symbol means there are generally two solutions or roots for \( x \).
Polynomial Inequality
Polynomial inequalities involve expressions where a polynomial is set to be greater than, less than, or equal to another expression.
In essence, they compare the size of the polynomial to a number or another polynomial.
Here's how to approach them:
In essence, they compare the size of the polynomial to a number or another polynomial.
Here's how to approach them:
- Identify the polynomial. In our problem, it's \( 16x^4 - 40x^2 + 9 \leq 0 \).
- Factor the polynomial, if possible, to find its roots. These roots are the critical points where the polynomial may change its sign.
- Determine the intervals created by these roots on the number line.
- Test each interval to check whether the inequality holds true. You do this by substituting a number from the interval into the original polynomial.
Factorization
Factorization is the mathematical process of breaking down a complex expression into simpler components, called factors, that when multiplied together give the original expression.
It's like finding the ingredients used in a recipe by looking at the finished dish.
Here鈥檚 an easy way to grasp factorization:
It's like finding the ingredients used in a recipe by looking at the finished dish.
Here鈥檚 an easy way to grasp factorization:
- Start by identifying patterns, such as common factors, trinomials, or special products like difference of squares.
- The goal is often to rewrite a polynomial so that it's easier to work with, especially in solving equations or inequalities.
- In expressions like \( 16u^2 - 40u + 9 \), we use the quadratic formula to find roots and then express it as \((4u-3)^2\), indicating the roots and simplifying further calculations.
