Question

Solve the inequality. Then graph the solution set on the real number line. \(x^{2}-6 x+9<16\)

Short answer

The solution to the inequality \(x^{2}-6 x+9<16\) is \(-1 < x < 7\)

Step-by-step solution

Simplify the inequality

Subtract 16 from both sides of the inequality: \(x^{2}-6x+9-16<0\). This simplifies to \(x^{2}-6x-7<0\).

Solve the quadratic equation

Set the simplified equation equal to zero and solve for x: \(x^{2}-6x-7=0\). To solve this quadratic equation in standard form \( ax^{2}+bx+c=0 \), use the quadratic formula \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \). Substituting a=1, b=-6, and c=-7 into the formula, you get solutions \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{6\pm\sqrt{(-6)^{2}-4\cdot1\cdot(-7)}}{2\cdot1}=\frac{6\pm\sqrt{36+28}}{2}=\frac{6\pm\sqrt{64}}{2}=\frac{6\pm8}{2}=7, -1 \)

Determine the intervals of x that satisfy the inequality

Plugging the roots \(x = -1\) and \(x = 7\) into the inequality, we can analyze it in the intervals \(-\infty < x < -1\), \(-1 < x < 7\), and \(7 < x < \infty\). It turns out that the inequality is true for \(-1 < x < 7\).

Plot the solution set in the number line

Draw a number line, mark the points x = -1 and x = 7 on the line. The solution for this inequality excludes the roots, so make an open circle at x = -1 and x = 7, then shade the region between these two points which represents \(-1 < x < 7\).

Key concepts

Quadratic Formula

The quadratic formula is an essential tool in algebra for solving quadratic equations. A quadratic equation is in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The quadratic formula is represented as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows you to find the values of \( x \) by substituting the coefficients \( a \), \( b \), and \( c \) from the equation directly.

• Start by calculating the discriminant, \( b^2 - 4ac \). The discriminant helps in understanding the nature of the roots.
• If the discriminant is positive, you will have two real distinct roots.
• If it's zero, you will have exactly one real root, also known as a repeated root.
• If the discriminant is negative, the equation has no real solutions but two complex ones.

The use of the quadratic formula makes solving messy equations straightforward, saving time and reducing error.

Number Line

A number line is a simple yet powerful tool used to visually represent numbers in their order from smallest to largest. It can include all types of numbers such as integers, fractions, and decimals. When graphing inequalities, we often use the number line to show where the inequality is true. For instance, to graph the solution set of the inequality \(-1 < x < 7\):

• First, draw a horizontal line.
• Mark significant points, in this case, \( x = -1 \) and \( x = 7 \).
• Use open circles at these points to indicate that these values are not included in the solution set.
• Shade the region between these two points to depict the continuous range of solutions.

A number line thus helps in clearly visualizing the intervals where an inequality holds true, enhancing understanding through graphical representation.

Solution Set

The solution set of an inequality comprises all possible values that satisfy the given inequality. For the quadratic inequality \( x^2 - 6x - 7 < 0 \), determining the solution involves several steps:

• Solve the corresponding equation \( x^2 - 6x - 7 = 0 \) to find critical points.
• The roots, or solutions, \( x = -1 \) and \( x = 7 \) divide the number line into intervals.
• Check each interval to see where the inequality holds true.
• The solution \(-1 < x < 7\) means all real numbers between -1 and 7 satisfy the inequality.

It's important to remember whether the inequality is strict (\(<\) or \(>\)) or inclusive (\(\leq\) or \(\geq\)), which affects whether endpoints are included in the solution set.

Real Numbers

Real numbers encompass all the numbers that can be found on the number line. This includes rational numbers (such as integers and fractions) and irrational numbers (like \( \sqrt{2} \) and \( \pi \)). They are fundamental in solving and expressing the solutions of inequalities and equations.When solving an inequality and expressing its solution set, as in \(-1 < x < 7\), we refer to real numbers. This acknowledges that any value between -1 and 7, including decimals and irrational numbers, are valid solutions.Real numbers have several critical properties that make them incredibly useful:

• They are dense, which means between any two real numbers, there exists another real number.
• They can be added, subtracted, multiplied, and divided (except by zero) while staying within the set of real numbers.
• They are ordered, meaning they can be placed in a meaningful sequence from least to greatest.

Understanding real numbers is crucial because they are the basis for most of our calculations and truly represent the 'real' world.