Question

If \(x^{2}-4 x+3>0\) and \(x^{2}-6 x+8<0\), then (1) \(x>3\) (2) \(x<4\) (3) \(3

Short answer

Answer: The correct range for x is \(3 < x < 4\).

Step-by-step solution

Solve the first inequality

Rewrite the inequality \(x^2 - 4x + 3 > 0\) as a quadratic equation. \((x - 1)(x - 3) > 0\) This means x should be either greater than 3 or less than 1 for this quadratic to be positive. So, the solution to the first inequality is: \(x < 1\) or \(x > 3\)

Solve the second inequality

Rewrite the inequality \(x^2 - 6x + 8 < 0\) as a quadratic equation. \((x - 2)(x - 4) < 0\) This means x should be between 2 and 4 for this quadratic to be negative. So, the solution to the second inequality is: \(2 < x < 4\)

Find the intersection of the solutions

To find the correct range for x, we need to find the intersection of the solutions of the two inequalities. This means we're looking for values of x that satisfy both conditions, those found in Steps 1 and 2. From the solutions of the two inequalities, we can see that the intersection of those ranges is: \(3 < x < 4\) Hence, the correct range for x is (3) \(3 < x < 4\).

Key concepts

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the range of values for the variable that will make the inequality true. Unlike quadratic equations, which equal a value, inequalities suggest a relationship where one side is either greater than or less than the other side. To solve a quadratic inequality like \(x^2 - 4x + 3 > 0\), we can factor the quadratic expression, if possible, and then determine the intervals based on the roots.

For example, if a factored form yields \((x - 1)(x - 3) > 0\), we understand that the solutions are intervals where the product of the factors is positive. By using a sign chart or analyzing the factor's signs, we discern that \(x < 1\) or \(x > 3\) are the intervals where the inequality holds true. These intervals are where the original quadratic inequality is greater than zero.

Quadratic Equations

Quadratic equations come in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. To solve such equations, we often start by factoring the quadratic expression into the product of two binomials, if possible. Alternatively, we can use the quadratic formula or complete the square.

For instance, the quadratic equation created from the inequality \(x^2 - 6x + 8 = 0\) can be factored to \((x - 2)(x - 4) = 0\). The solutions, called 'roots', are the values where the product equals zero, in this case, \(x = 2\) and \(x = 4\). These roots are critical in determining the intervals where the original inequality holds true.

Inequality Intervals

Inequality intervals represent the range of values that satisfy a given inequality. When we factor a quadratic inequality, the roots divide the number line into distinct intervals. For each interval, we must determine whether the inequality holds true by testing values or analyzing the signs of the factors.

The inequality \(x^2 - 6x + 8 < 0\), factored as \((x - 2)(x - 4) < 0\), creates intervals that we need to consider: \(x < 2\), \(2 < x < 4\), and \(x > 4\). By testing points or using sign analysis, we find that the original inequality is satisfied for the interval \(2 < x < 4\) only, as the product of the factors within this interval is negative.

Intersection of Solutions

When solving systems of inequalities, we find the intersection of solutions, which are the values that satisfy all inequalities in the system. The intersection can be visualized by overlapping the intervals on a number line and observing where they meet.

Given two inequalities like \(x^2 - 4x + 3 > 0\) and \(x^2 - 6x + 8 < 0\), we find their solution intervals are \(x < 1\) or \(x > 3\), and \(2 < x < 4\), respectively. The intersection of these intervals is where both conditions are true. In this case, the values of \(x\) that lie within the overlapping regions are \(3 < x < 4\). Therefore, the intersection of solutions for these two inequalities is the interval where both are valid, leading us to conclude that the correct response to the exercise is \(3 < x < 4\).