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Chapter 6: Problem 27

Solve each inequality. $$8 x^{2}+22 x+5 \geq 0$$

Short Answer

Expert verified
Solution is \((-\infty, -\frac{5}{2}] \cup [-\frac{1}{2}, \infty)\).

Step by step solution

01

Identify the form of the inequality

The given inequality is \( 8x^2 + 22x + 5 \geq 0 \). Notice it is a quadratic inequality because it contains an \(x^2\) term.
02

Set up the corresponding quadratic equation

We first find the solutions to the quadratic equation by setting \( 8x^2 + 22x + 5 = 0 \). Solving this equation will help us find the critical points.
03

Use the quadratic formula

For \( ax^2 + bx + c = 0 \), the solutions are given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here \( a = 8 \), \( b = 22 \), and \( c = 5 \). Calculating, we get \( x = \frac{-22 \pm \sqrt{22^2 - 4 \cdot 8 \cdot 5}}{2 \cdot 8} \).
04

Simplify the discriminant

Calculate \( 22^2 - 4 \cdot 8 \cdot 5 = 484 - 160 = 324 \). So, the discriminant, \( \sqrt{324} = 18 \).
05

Solve for x

Substitute back into the formula: \( x = \frac{-22 \pm 18}{16} \). This gives two solutions: \( x = \frac{-22 + 18}{16} = -\frac{1}{2} \) and \( x = \frac{-22 - 18}{16} = -\frac{5}{2} \).
06

Determine intervals for solution

The critical points divide the x-axis into three intervals: \((-\infty, -\frac{5}{2})\), \([-\frac{5}{2}, -\frac{1}{2}]\), and \((-\frac{1}{2}, \infty)\). Use test points from each interval to determine where the inequality \( 8x^2 + 22x + 5 \geq 0 \) holds.
07

Test the intervals

Choose a test point in \((-\infty, -\frac{5}{2})\), say \( x = -3 \): \( 8(-3)^2 + 22(-3) + 5 = 72 - 66 + 5 = 11 \geq 0 \). In \([-\frac{5}{2}, -\frac{1}{2}]\), choose \( x = -1 \): \( 8(-1)^2 + 22(-1) + 5 = 8 - 22 + 5 = -9 ot\geq 0 \). In \((-\frac{1}{2}, \infty)\), choose \( x = 0 \): \( 8(0)^2 + 22(0) + 5 = 5 \geq 0 \).
08

Conclude the solution

The inequality is true for the intervals \((-\infty, -\frac{5}{2}]\) and \([-\frac{1}{2}, \infty)\). The solution of the inequality is thus \((-\infty, -\frac{5}{2}] \cup [-\frac{1}{2}, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equation
A quadratic equation is a type of polynomial equation that follows the form: \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The variable \( x \) represents an unknown value we are solving for. Quadratic equations are called 'quadratic' because the highest degree of the variable \( x \) is 2, as indicated by the square term \( x^2 \). These equations often produce two solutions or roots, as opposed to linear equations which have only one. They create a symmetric parabolic shape when graphed on a coordinate plane. Quadratic equations are foundational in algebra and have numerous applications in science, engineering, and finance.
Quadratic Formula
The quadratic formula is a universal tool to solve any quadratic equation. This formula is especially useful when factoring is difficult or impossible. The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this formula:
  • \( b^2 - 4ac \) is the discriminant, which determines the nature of the roots.
  • The "\( \pm \)" symbol means there will be two solutions: one with addition and one with subtraction.
The quadratic formula provides the exact solutions to the equation by considering all possible roots based on the coefficients \( a \), \( b \), and \( c \). It is derived from completing the square method and works for all types of quadratic equations including those with imaginary roots.
Inequality Solution Intervals
When solving a quadratic inequality, such as \( ax^2 + bx + c \geq 0 \), the goal is to identify the intervals on the \( x \)-axis where the inequality holds true. The solution process involves several steps:
  • First, solve the corresponding quadratic equation \( ax^2 + bx + c = 0 \) to find the critical points or roots.
  • These roots divide the \( x \)-axis into several intervals.
  • Test a point within each of these intervals to determine whether the inequality is satisfied in that interval.
The roots themselves are often part of the solution, particularly if the inequality involves a \( \geq \) or \( \leq \) sign. The resulting intervals are expressed in interval notation, combining them to form the full solution set for the inequality.
Discriminant Calculation
The discriminant is a key component in understanding the nature of the roots of a quadratic equation. It is represented by the expression \( b^2 - 4ac \) within the quadratic formula.Depending on the value of the discriminant:
  • Positive discriminant: indicates two distinct real roots.
  • Zero discriminant: implies exactly one real root (a repeated root).
  • Negative discriminant: results in two complex conjugate roots (no real roots).
Calculating the discriminant is crucial for determining the intervals over which a quadratic inequality holds because it helps pinpoint the locations of the roots on a number line. In the given problem, the discriminant calculation showed a positive value (324), which indicated that the quadratic equation has two real and distinct solutions, critical for determining solution intervals for the inequality.

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Most popular questions from this chapter

The product \((x-2)(x+3)\) is positive if both factors are negative or if both factors are positive. Therefore, we can solve \((x-2)(x+3)>0\) as follows: $$ \begin{gathered} (x-2<0 \text { and } x+3<0) \text { or }(x-2>0 \text { and } x+3>0) \\ (x<2 \text { and } x<-3) \text { or }(x>2 \text { and } x>-3) \\ x<-3 \text { or } x>2 \end{gathered} $$ The solution set is \((-\infty,-3) \cup(2, \infty)\). Use this type of analysis to solve each of the following. (a) \((x-2)(x+7)>0\) (b) \((x-3)(x+9) \geq 0\) (c) \((x+1)(x-6) \leq 0\) (d) \((x+4)(x-8)<0\) (e) \(\frac{x+4}{x-7}>0\) (f) \(\frac{x-5}{x+8} \leq 0\)

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