Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 42
Solve the given nonlinear inequality. Write the solution set using interval notation. Graph the solution set. $$ 4 x^{2}>9 x+9 $$
Short Answer
Expert verified
The solution is \((-\infty, -\frac{3}{4}) \cup (3, \infty)\).
Step by step solution
01
Rewrite the Inequality
First, rewrite the given inequality into a standard form. Move all terms to one side of the inequality: \[ 4x^2 - 9x - 9 > 0 \] This will help us identify critical points where the inequality may change signs.
02
Find Critical Points
To find the critical points, solve the equation:\[ 4x^2 - 9x - 9 = 0 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 4 \), \( b = -9 \), and \( c = -9 \):\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4} \]Calculate the discriminant and solve for \( x \):\[ x = \frac{9 \pm \sqrt{81 + 144}}{8} = \frac{9 \pm \sqrt{225}}{8} \]\[ x = \frac{9 \pm 15}{8} \]This gives solutions \( x = 3 \) and \( x = -\frac{3}{4} \).
03
Test Intervals
With critical points \( x = -\frac{3}{4} \) and \( x = 3 \), test intervals created by these points, i.e., \(( -\infty, -\frac{3}{4} )\), \((-\frac{3}{4}, 3)\), and \((3, \infty)\).Select test points from these intervals:- For \(( -\infty, -\frac{3}{4} )\), test \( x = -1 \).- For \((-\frac{3}{4}, 3)\), test \( x = 0 \).- For \((3, \infty)\), test \( x = 4 \).Substitute these test points into the expression \( 4x^2 - 9x - 9 \) to check if the result is positive or negative.
04
Evaluate Test Points
Evaluate the sign of the expression for each test point:- For \( x = -1 \), \( 4(-1)^2 - 9(-1) - 9 = 4 + 9 - 9 = 4 \). Positive.- For \( x = 0 \), \( 4(0)^2 - 9(0) - 9 = -9 \). Negative.- For \( x = 4 \), \( 4(4)^2 - 9(4) - 9 = 64 - 36 - 9 = 19 \). Positive.Thus, the inequality \( 4x^2 - 9x - 9 > 0 \) holds for \( x \) in the intervals \(( -\infty, -\frac{3}{4} )\) and \((3, \infty)\).
05
Write Solution in Interval Notation
Considering where the inequality is true \( 4x^2 - 9x - 9 > 0 \), write these in interval notation:\[ (-\infty, -\frac{3}{4}) \cup (3, \infty) \]
06
Graph the Solution Set
Graph the solution set on a number line. Shade the regions \(( -\infty, -\frac{3}{4} )\) as well as \((3, \infty)\), and use open circles at \( x = -\frac{3}{4} \) and \( x = 3 \) since these points are not included in the solutions.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Inequalities
In mathematics, a quadratic inequality involves an expression of degree two and a relational symbol, typically ">" or "<". These inequalities appear in the form \( ax^2 + bx + c > 0 \) or \( ax^2 + bx + c < 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The inequality signifies that the quadratic expression is either greater than or less than zero.
To solve quadratic inequalities, follow these steps:
To solve quadratic inequalities, follow these steps:
- Rewrite the inequality in the standard form, moving all terms to one side of the inequality.
- Solve the corresponding quadratic equation \( ax^2 + bx + c = 0 \) to find the critical points.
- Test intervals created by these critical points to determine where the inequality holds true.
Interval Notation
Interval notation is a numerical expression of a set of numbers lying between two endpoints. It is widely used to represent the solution sets of inequalities clearly and concisely.
Here are key aspects of interval notation:
Here are key aspects of interval notation:
- Use brackets "[]" to represent inclusive boundaries where endpoints are part of the set.
- Use parentheses "()" to represent exclusive boundaries where endpoints are not part of the set.
- The symbol "\( \cup \)" denotes a union, signaling a combination of two intervals.
Graphing Inequalities
Graphing inequalities involves visually representing the solution set of an inequality on a number line. This process helps in understanding which parts of the number line satisfy the inequality.
Follow these steps to graph inequalities:
Follow these steps to graph inequalities:
- Identify the critical points and mark them on the number line.
- Use open circles for points that are not included in the solution and closed circles for included points.
- Shade the regions on the number line corresponding to intervals where the inequality holds true.
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of a quadratic equation. It is applicable when expressions cannot be easily factored.
The formula itself is given as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The term under the square root \( b^2 - 4ac \) is called the discriminant and determines the nature of the roots:
The formula itself is given as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The term under the square root \( b^2 - 4ac \) is called the discriminant and determines the nature of the roots:
- If the discriminant is positive, two distinct real roots exist.
- If the discriminant is zero, there is exactly one real root.
- If the discriminant is negative, the quadratic has no real roots.
