Question

Solve the inequality. Then graph the solution set on the real number line. \(x^{2} \leq 9\)

Short answer

The solution to the inequality \(x^{2} \leq 9\) is \(-3 \leq x \leq 3\). On the real number line, this is represented by a closed segment from -3 to 3.

Step-by-step solution

Identify Inequality Type

First, it's essential to note that the given equation is a quadratic inequality as it is based on a quadratic function \(x^{2}\). The inequality symbol used is \(\leq\) which means our solution will include values where \(x^{2}\) equals 9 and also where \(x^{2}\) is less than 9.

Squareroot Both Sides

To solve the inequality, we equivalent square root both sides of the inequality. Remember that when you square root both sides include both positive and negative roots. This leads to \(-3 \leq x \leq 3\). The solution to this inequality includes all x-values that fulfill this condition.

Graph

On the number-line, locate and mark a point each for -3 and 3. Since the inequality sign includes equality as well (\(\leq\)), both -3 and 3 are part of the solution. So the points will be closed (or filled). Draw a line segment connecting these two points to show all the numbers in between -3 and 3 are part of the solution.

Key concepts

Solving Quadratic Inequalities

When tasked with solving a quadratic inequality like \(x^{2} \leq 9\), the process is slightly different from solving a simple equation. A quadratic inequality is not just about finding equal values but determining a range of x-values that make the inequality true.

For instance, breaking down our example, \(x^{2} \leq 9\) suggests that we are finding all the x-values where the square of x is at most 9. This includes not only the points where \(x^2\) equals to 9 but also where it's less. The solution to this will be a set of x-values rather than a single number. When working with inequalities, it's important to pay attention to the inequality sign. Here, \(\leq\) indicates that our solution includes the points where the inequality is equal and where it is less than 9. This distinction is crucial when interpreting and graphing your solution.

Number Line Graphing

Graphing the solutions to inequalities on a number line is an excellent way to visually represent the range of possible values that satisfy the inequality. With our example, \(-3 \leq x \leq 3\), we use the number line to show that every point from -3 to 3, including the endpoints, is part of the solution.

Here's how it's done: find the points that correspond to the boundaries of our solution, in this case, -3 and 3. These points are marked on the number line, usually with a dot or a circle. Because our inequality includes the endpoints (indicated by the \(\leq\) sign), we'll use closed dots to signify that -3 and 3 themselves are solutions. Then we draw a solid line or bar to connect these dots, showing that all the numbers in-between are also solutions. This graphical representation is not only intuitive but also provides immediate visual insight into the nature of the inequality solution.

Understanding Quadratic Functions

At the heart of our example inequality, \(x^{2} \leq 9\), is a quadratic function, which is a type of polynomial where the highest exponent of the variable is two, represented generally as \(ax^{2} + bx + c\).

The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient of the \(x^{2}\) term. In the context of inequalities, it's essential to understand the shape of the parabola because it tells us about the range of values for which the inequality can hold true. For example, if a quadratic function opens upwards and we're dealing with a '\(\leq\)' inequality sign, the solution set includes values below the parabola. Conversely, if it opens downwards, the solution set involves values above it. However, in our simplified inequality, the quadratic is \(x^{2}\), which represents a parabola that opens upward and is symmetrical about the y-axis.

The Square Root Method

One efficient way to solve quadratic inequalities, like our example \(x^{2} \leq 9\), is by utilizing the square root method. This method involves taking the square root of both sides of the inequality, but here's where special care is needed.

When square rooting as part of solving an inequality, we have to consider both the positive and negative square roots of the number on the opposite side of the inequation. That's because squaring either a positive or negative number yields a positive result. For our equation, taking the square root of 9 gives us 3, but we must remember to include -3 as a solution too due to the nature of squares. Therefore, we end up with a solution set that reads \(-3 \leq x \leq 3\), encompassing all x-values within and including the endpoints of this range.