Chapter 1: Problem 2
Find the test intervals of the inequality. \(x^{2}-6 x+8>0\)
Short Answer
Expert verified
The test intervals of the inequality \(x^{2}-6x+8 > 0\) are \((4, \infty)\).
Step by step solution
01
Find the Roots
Here, the first task is to find the roots of the quadratic equation \(x^{2}-6x+8=0\). This can be achieved by factorising the quadratic (if possible) or by using the formula for roots of a quadratic equation which is \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where a, b, c are coefficients of the quadratic equation \(ax^{2} + bx + c\). After solving, you will find that the roots of the equation are \(x = 2\) and \(x = 4\).
02
Find the Intervals
Now that you have the roots, you can use these to create three test intervals. The intervals are \((- \infty, 2)\), \((2, 4)\), and \((4, \infty)\).
03
Test the Intervals
Choose a test number from each interval and evaluate it in the inequality \(x^{2}-6x+8 > 0\). Let's take 1, 3, and 5 as test numbers from the respective intervals. You will get these values after plugging the numbers into the inequality 3(-ve), -1(-ve) and 3(+ve) respectively. The inequality holds true for intervals where the value is +ve.
04
Write down the Solution
The evaluation from Step 3 shows that the solution to the inequality is the interval \((4, \infty)\). The inequality \(x^{2}-6x+8 > 0\) holds true for this interval.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation Roots
Understanding the roots of a quadratic equation is the cornerstone to solving quadratic inequalities.
The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are coefficients. To find the roots, that is, the values of \( x \) which satisfy the equation, you can factorise the quadratic or use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
For instance, the equation from our exercise \( x^2 - 6x + 8 = 0 \) yields two roots, \( x = 2 \) and \( x = 4 \) when factorised. These roots split the number line into distinct intervals which are then used to find where the inequality holds true. Being able to accurately find and interpret these roots is essential for progressing through any quadratic inequalities problems.
The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are coefficients. To find the roots, that is, the values of \( x \) which satisfy the equation, you can factorise the quadratic or use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
For instance, the equation from our exercise \( x^2 - 6x + 8 = 0 \) yields two roots, \( x = 2 \) and \( x = 4 \) when factorised. These roots split the number line into distinct intervals which are then used to find where the inequality holds true. Being able to accurately find and interpret these roots is essential for progressing through any quadratic inequalities problems.
Test Intervals
Once the roots are identified, the next step is creating 'test intervals' on the number line.
These intervals are derived from the roots and represent the ranges within which we will evaluate the inequality.
In our example, the roots of \( x = 2 \) and \( x = 4 \) divide the number line into three intervals: \( (-\infty, 2) \) (all values less than 2), \( (2, 4) \) (values between 2 and 4), and \( (4, \infty) \) (all values greater than 4). Evaluating the inequality within these intervals will determine the solution set of the original inequality. This method is practical as it narrows down the infinite number of possibilities to just a few intervals to consider.
These intervals are derived from the roots and represent the ranges within which we will evaluate the inequality.
In our example, the roots of \( x = 2 \) and \( x = 4 \) divide the number line into three intervals: \( (-\infty, 2) \) (all values less than 2), \( (2, 4) \) (values between 2 and 4), and \( (4, \infty) \) (all values greater than 4). Evaluating the inequality within these intervals will determine the solution set of the original inequality. This method is practical as it narrows down the infinite number of possibilities to just a few intervals to consider.
Inequality Solutions
Determining the solution set for a quadratic inequality involves evaluating the inequality with numbers from the test intervals identified.
Pick a number from each interval and plug it into the original inequality.
For examle, we may take \( 1 \) from \( (-\infty, 2) \) interval, \( 3 \) from \( (2, 4) \) interval, and \( 5 \) from \( (4, \infty) \). In our original equation \( x^2 - 6x + 8 > 0 \) when these numbers are substituted, we discover that only the test number from the \( (4, \infty) \) interval results in a positive value, satisfying the inequality.
The solution set of the inequality is thus all values within the interval that yield a positive result upon substitution, which, in our case, is \( (4, \infty) \).
Pick a number from each interval and plug it into the original inequality.
For examle, we may take \( 1 \) from \( (-\infty, 2) \) interval, \( 3 \) from \( (2, 4) \) interval, and \( 5 \) from \( (4, \infty) \). In our original equation \( x^2 - 6x + 8 > 0 \) when these numbers are substituted, we discover that only the test number from the \( (4, \infty) \) interval results in a positive value, satisfying the inequality.
The solution set of the inequality is thus all values within the interval that yield a positive result upon substitution, which, in our case, is \( (4, \infty) \).
Factorising Quadratics
Factorising quadratics is a technique used to express the quadratic equation as a product of two binomials.
This method is useful for finding the roots of the equation, making it easier to solve quadratic inequalities.
The equation \( x^2 - 6x + 8 \) can be factorised as \( (x - 2)(x - 4) = 0 \). When these factors are equal to zero, it gives us the roots. This process simplifies the identification of intervals since the roots are the points where the sign of the inequality can potentially change.
Understanding how to factorise quadratics is crucial for breaking down and solving these kinds of inequality problems more efficiently and is often the first step you'll take.
This method is useful for finding the roots of the equation, making it easier to solve quadratic inequalities.
The equation \( x^2 - 6x + 8 \) can be factorised as \( (x - 2)(x - 4) = 0 \). When these factors are equal to zero, it gives us the roots. This process simplifies the identification of intervals since the roots are the points where the sign of the inequality can potentially change.
Understanding how to factorise quadratics is crucial for breaking down and solving these kinds of inequality problems more efficiently and is often the first step you'll take.
