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Magazine Chapter 3: Problem 88
Solve each inequality by using the method of your choice. State the solution set in interval notation and graph it. $$9 s^{2}+6 s+1 \geq 0$$
Short Answer
Expert verified
The solution is all real numbers $$(-\infty, \infty)$$
Step by step solution
01
- Recognize the quadratic inequality
The given inequality is a quadratic inequality: $$9s^2 + 6s + 1 \geq 0$$
02
- Express in standard form
The inequality is already in the standard quadratic form: $$as^2 + bs + c \geq 0$$ where \ a = 9, b = 6, c = 1$$.
03
- Find the discriminant
The discriminant of a quadratic equation $$ax^2 + bx + c \ is given by \Delta = b^2 - 4ac$$. calculate the discriminant: \begin{align*} \Delta &= (6)^2 - 4(9)(1) \ &= 36 - 36 \ &= 0 \end{align*}
04
- Solve the equation for s
Since the discriminant is zero, the quadratic equation has exactly one real root. Solve for s using $$s = \frac{-b}{2a}$$: \begin{align*} \ s &= \frac{-6}{2(9)} \ &= \frac{-6}{18} \ &= -\frac{1}{3} \end{align*}
05
- Analyze the quadratic function's sign
The quadratic function $$y = 9s^2 + 6s + 1 \geq 0$$ represents a parabola that opens upwards (since a > 0). Since the only root is $$s=-\frac{1}{3}$$, the parabola does not cross the x-axis and only touches the x-axis at this point. Thus, $$y = 9s^2 + 6s + 1 \geq 0$$ for all real $$s$$.
06
- State the solution set in interval notation
Since the quadratic expression is always greater than or equal to 0, the solution set is all real numbers: $$(-\infty, \infty)$$.
07
- Graph the solution on the number line
Draw a number line covering $$s$$ values from $$-\infty$$ to $$\infty$$. Mark $$-\frac{1}{3}$$ and shade the entire line to indicate that all the values are solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
The Discriminant Explained
To solve the quadratic inequality, you need to understand a critical component: the discriminant. The discriminant is a part of the quadratic formula, given by the expression \(鈭� = b^2 - 4ac\). It helps us determine the nature of the roots of the quadratic equation. The discriminant provides essential information:
- When 鈭� > 0, there are two distinct real roots.
- When 鈭� = 0, there is exactly one real root.
- When 鈭� < 0, there are no real roots, only complex roots.
Understanding Parabolas
A quadratic function like 9s^2 + 6s + 1 forms a 'U'-shaped curve called a parabola. The coefficient of the s^2 term (a = 9) determines if the parabola opens upwards (a > 0) or downwards (a < 0).
In this case, since a = 9, our parabola opens upwards. The vertex of the parabola is the highest or lowest point. For inequalities, solving revolves around when the parabola is above or below the x-axis.
Given our problem, the vertex of the parabola touches the x-axis at one point, s = -1/3, and does not cross it. Thus, the values of s that satisfy our inequality will be affected by this vertex.
In this case, since a = 9, our parabola opens upwards. The vertex of the parabola is the highest or lowest point. For inequalities, solving revolves around when the parabola is above or below the x-axis.
Given our problem, the vertex of the parabola touches the x-axis at one point, s = -1/3, and does not cross it. Thus, the values of s that satisfy our inequality will be affected by this vertex.
Interval Notation
Interval notation is a way of writing the set of all solutions to an inequality. It uses brackets and parentheses to describe intervals on the number line. Here's a quick guide:
For our quadratic inequality, since the expression is always greater than or equal to zero, the solution in interval notation is (-鈭�, 鈭�). This indicates that any real number s is a solution to the inequality.
- (a, b) means all numbers between a and b, not including a and b.
- [a, b] means all numbers between a and b, including a and b.
- (a, b] means all numbers between a and b, not including a, but including b.
- Use -鈭� for negative infinity and 鈭� for positive infinity.
For our quadratic inequality, since the expression is always greater than or equal to zero, the solution in interval notation is (-鈭�, 鈭�). This indicates that any real number s is a solution to the inequality.
Solving Quadratic Inequalities
Solving quadratic inequalities involves a few standard steps:
In our example, we followed these steps and determined that 9s^2 + 6s + 1 is always non-negative. Hence, the solution to the inequality 9s^2 + 6s + 1 鈮� 0 is all real numbers, expressed as (-鈭�, 鈭�). By understanding these steps, you can confidently solve any quadratic inequality you encounter.
- Convert the inequality to its standard form, if necessary.
- Find the roots of the corresponding quadratic equation by calculating the discriminant.
- Determine where the quadratic function is positive or negative by analyzing the graph of the parabola.
- Write the solution in interval notation.
In our example, we followed these steps and determined that 9s^2 + 6s + 1 is always non-negative. Hence, the solution to the inequality 9s^2 + 6s + 1 鈮� 0 is all real numbers, expressed as (-鈭�, 鈭�). By understanding these steps, you can confidently solve any quadratic inequality you encounter.
