Problem 30 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ x^{2}<25 $$

Step-by-Step Solution

Answer

The solution set is \((-5, 5)\).

Step 1: Understand the Inequality

The given inequality is \( x^2 < 25 \). We need to find all the values of \( x \) where \( x^2 \) is less than 25. This is a quadratic inequality.

Step 2: Solve the Corresponding Equation

First, find the points where the function is equal to 25. Solve the equation \( x^2 = 25 \). By taking the square root of both sides, we have \( x = 5 \) and \( x = -5 \). These are the boundary points that separate solutions from non-solutions.

Step 3: Consider the Test Intervals

The solutions of the equation \( x^2 = 25 \) divide the number line into three intervals: \((-\infty, -5)\), \((-5, 5)\), and \((5, \infty)\). We need to test a point in each interval to determine where \( x^2 < 25 \).

Step 4: Test the Intervals

Choose a test point from each interval:- For \((-\infty, -5)\), use \(x = -6\). \( (-6)^2 = 36 \), which is not less than 25.- For \((-5, 5)\), use \(x = 0\). \( 0^2 = 0 \), which is less than 25.- For \((5, \infty)\), use \(x = 6\). \( 6^2 = 36 \), which is not less than 25.The inequality holds true for \( x \) values between -5 and 5.

Step 5: Write the Solution in Interval Notation

From the testing, the solution to the inequality \( x^2 < 25 \) is the interval \((-5, 5)\). This interval includes all the \( x \) values that satisfy the inequality.

Key Concepts

Interval Notation

Interval notation is a way of representing a range of values that satisfy a particular condition or inequality. It provides a concise way to express the solution set for inequalities.
For the inequality solution of quadratic inequalities like \( x^2 < 25 \), we determine the range of \( x \) values that make the inequality true. In this case, as per step 5, the values of \( x \) that satisfy \( x^2 < 25 \) are all between -5 and 5. Thus, the interval is written as

• \((-5, 5)\)

In this interval notation:

• The parentheses \( () \) indicate that the endpoints -5 and 5 are not included in the solution, meaning \( x \) can be greater than -5 but less than 5.

This specific notation represents all real numbers between -5 and 5 without including -5 and 5 themselves. It is important in mathematics as it offers simplicity and clarity when communicating ranges of numbers.

Boundary Points

Boundary points are critical in solving quadratic inequalities as they separate regions where the inequality holds true from those where it doesn't. These points occur where the quadratic expression equals the constant on the other side of the inequality.
In our example, for \( x^2 < 25 \), we first convert the inequality to an equation: \( x^2 = 25 \). Solving this gives us the boundary points \( x = 5 \) and \( x = -5 \). These points do not satisfy the inequality \( x^2 < 25 \) but are crucial as they mark transitions between different intervals.
To determine where the inequality is true, we examine what happens around these boundary points by testing intervals divided by these points. But remember, since the inequality is \( < \) and not \( \leq \), points like -5 and 5 are not included in the solution set.

Test Intervals

Test intervals are used to check which segments of the number line satisfy a given inequality. For a quadratic inequality like \( x^2 < 25 \), once you have identified the boundary points, these divide the number line into distinct regions, or intervals.

For \( x^2 < 25 \), the boundary points -5 and 5 separate the real number line into three intervals:

• \((-\infty, -5)\)
• \((-5, 5)\)
• \((5, \infty)\)

We select a test point from each interval to determine if the inequality is true in that interval:

• For \((-\infty, -5)\), using a test point like \( x = -6 \), gives \((-6)^2 = 36\), which is not less than 25, so this interval does not satisfy the inequality.
• For \((-5, 5)\), using \( x = 0 \), gives \(0^2 = 0\), which is less than 25, meaning this interval satisfies the inequality.
• For \((5, \infty)\), using \( x = 6 \), gives \(6^2 = 36\), which is also not less than 25, so this interval does not satisfy the inequality.

Test intervals are essential for confirming and understanding which parts of the number line solve the inequality. This method allows for accurate identification of the solution set when dealing with inequalities.