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Chapter 3: Problem 2

Explain how to solve \(x^2+6 x-8<0\) using algebraic methods and using graphs.

Short Answer

Expert verified
The algebraic solution to the inequality \(x^2+6 x-8<0\) is \(x<-8\,\, or\,\, -8<x<1\). Based on the graph of the parabola \(y=x^2+6x-8\), the solution set of the inequality corresponds to the x-values of the region below the x-axis.

Step by step solution

01

Solve the Equation

First, solve \(x^2+6 x-8=0\) . This involves factoring the quadratic expression into two binomial expressions. The quadratic equation equals zero when \(x = -8\) or \(x= 1\).
02

Construct the Number Line

Draw a number line and label the points -8 and 1. These will be the test points used to determine the solution set of the inequality.
03

Determine the Solution Set

Choose any number below -8, between -8 and 1, and above 1. Plug these values into the inequality and determine whether they render the inequality true or false.
04

Graph the Function

With the equation form, \(y = x^2+6 x-8\), plot a parabola. Identify where y crosses zero, where the solutions to the inequality lie.
05

Analyze the Graph

The areas below the x-axis where the solution of the inequality lies are reflected on the number line. These correspond to the values of x for which \(x^2+6 x-8<0\)

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Key Concepts

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

Factoring Quadratics
Solving a quadratic inequality like \(x^2+6x-8<0\) starts with factoring the quadratic equation. Factoring involves breaking down the expression into simpler binomials.
To do this, we first solve the equation \(x^2+6x-8=0\). The goal is to express it in the form \((x+a)(x+b)=0\). For our equation, factoring results in \((x+8)(x-1)=0\).
This tells us that the solutions where the quadratic equals zero are \(x = -8\) and \(x = 1\). These values are called the roots of the quadratic. They are critical in identifying the intervals for testing the inequality.
Number Line Analysis
A number line is a helpful visual tool for understanding the intervals of the solutions. After finding the roots at \(x = -8\) and \(x = 1\), these points divide the number line into three distinct regions.
To determine where the inequality \(x^2+6x-8<0\) holds true, select any number from each region. Plug this number into the inequality to test whether it makes the expression true.
  • Region 1: Pick a value less than -8, such as -9.
  • Region 2: Pick a value between -8 and 1, like 0.
  • Region 3: Pick a value greater than 1, for instance, 2.
If the inequality holds true in a region, it means the inequality is satisfied for all numbers within that segment of the number line.
Graphing Quadratic Functions
Graphing the quadratic function \(y = x^2+6x-8\) can provide a visual understanding of where the inequality is satisfied.
The graph of this function is a parabola, which opens upwards, as the coefficient of \(x^2\) is positive. The roots or x-intercepts at \(x = -8\) and \(x = 1\) are the points where the parabola crosses the x-axis.
To graph the function, plot these intercepts on the x-axis and sketch the parabola. The portions below the x-axis represent where the inequality \(x^2+6x-8<0\) is true, as these y-values are negative.
Parabolas and Inequalities
Understanding the connection between parabolas and inequalities is key to solving quadratic inequalities. When looking at a parabola like \(y=x^2+6x-8\), it is essential to note the regions relative to the x-axis.
The inequality \(x^2+6x-8<0\) is concerned with the parts of the graph where the parabola is below the x-axis, indicating negative y-values.
  • The graph dips below the x-axis between its x-intercepts \(-8\) and \(1\).
  • This interval \((-8, 1)\) on the number line shows where all solutions to the inequality lie.
This method highlights finding the solution set as the interval of x-values where the graph of the quadratic function remains below the x-axis.

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