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Magazine Chapter 1: Problem 27
In Exercises 11 through 32 , find the solution set of the given inequality and illustrate the solution on the real number $$ 4 x^{2}+9 x<9 $$
Short Answer
Expert verified
The solution set is \( (-3, \frac{3}{4}) \).
Step by step solution
01
- Rewrite the Inequality
Start by rewriting the inequality in the standard form: \[ 4x^2 + 9x - 9 < 0 \]
02
- Find the Roots of the Quadratic Equation
To determine where the inequality changes sign, solve the corresponding equation: \[ 4x^2 + 9x - 9 = 0 \] Use the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 4 \), \( b = 9 \), and \( c = -9 \).
03
- Calculate the Discriminant
Calculate the discriminant \( \Delta \): \[ \Delta = b^2 - 4ac \] \[ \Delta = 9^2 - 4(4)(-9) = 81 + 144 = 225 \]
04
- Find the Roots
Substitute \( \Delta = 225 \) back into the quadratic formula: \[ x = \frac{-9 \pm \sqrt{225}}{8} \] \[ x = \frac{-9 \pm 15}{8} \] So the roots are: \[ x_1 = \frac{6}{8} = \frac{3}{4} \] \[ x_2 = \frac{-24}{8} = -3 \]
05
- Determine the Intervals for Testing
The inequality \( 4x^2 + 9x - 9 < 0 \) changes sign at the roots. Check the inequality in the intervals: \[ (-\infty, -3), (-3, \frac{3}{4}), (\frac{3}{4}, \infty) \]
06
- Test Each Interval
Select a test point in each interval and determine if it satisfies the inequality. For example:1. For \( x = -4 \) in \((-\infty, -3)\): \[ 4(-4)^2 + 9(-4) - 9 = 64 - 36 - 9 = 19 > 0 \] Thus, \((-\infty, -3)\) is not part of the solution.2. For \( x = 0 \) in \((-3, \frac{3}{4})\): \[ 4(0)^2 + 9(0) - 9 = -9 < 0 \] Thus, \((-3, \frac{3}{4})\) is part of the solution.3. For \( x = 1 \) in \((\frac{3}{4}, \infty)\): \[ 4(1)^2 + 9(1) - 9 = 4 + 9 - 9 = 4 > 0 \] Thus, \((\frac{3}{4}, \infty)\) is not part of the solution.
07
- Write the Solution Set
Combine the results from testing the intervals. The inequality \( 4x^2 + 9x - 9 < 0 \) holds in the interval \[ (-3, \frac{3}{4}) \].
08
- Illustrate on the Real Number Line
Draw a number line and shade the interval \( (-3, \frac{3}{4}) \). Use open circles at \( -3 \) and \( \frac{3}{4} \) to indicate that these points are not included in the solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
To use the formula, identify the coefficients:
- The coefficient b in front of x.
- The constant term c.
Discriminant
Solution Interval
Next, select values from these intervals and plug them into the original inequality.
In our example:
- By selecting &150; -鈭�, -3), for x=-4\textThe expression, 4(-4)^2 + 9(-4) 鈥� 91) Both yields a positive value .
- Choose a number from (-3, \(\frac{3}{4}\)), for x=0 By plugging this value into the inequality we get,
- Choose a number from (鈭�, x=4
)Plugging in this vale to the inequality again results in positive 4>0. thus, this interval is not part of the solution So, the final range of z< (-3鈮�3,4)
After finding the intervals that satisfy the inequality, you can bsolve them.This set can be described using interval-notation.
Interval Notation
Interval notation is a concise way to describe intervals on the number line. It includes all the numbers between two endpoints.
To understand this better, let's break it down:
When we have a solution like (-3, 鈿(\frac{3}{4}\)), we write it as a shortend form showing that all numbers between -3 and \(\frac{3}{4}\) all belong to the solutions set.
In the interval I notation,
(use open parenthesis whenever an endpoint a< is not including. For closed point such as [including use
To understand this better, let's break it down:
When we have a solution like (-3, 鈿(\frac{3}{4}\)), we write it as a shortend form showing that all numbers between -3 and \(\frac{3}{4}\) all belong to the solutions set.
In the interval I notation,
(use open parenthesis whenever an endpoint a< is not including. For closed point such as [including use
In our example, the solution set is (<-3,3/4 Starting with -3 and end & usat the turning point of 戮)
Moreover, open circles are used to indicate that endpoints -3 &3,4x are not part of the point set .
This averted roots give us the complete soloption showing that that equation has -3<3/4 has 0 (solution denoting that equation>$ of solution should be used in that context completing the meaning will g example intervals.
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