Question

In Problems \(7-22,\) solve each inequality. 12\. \(x^{2}-1<0\)

Short answer

\(-1 < x < 1\)

Step-by-step solution

- Rewrite the Inequality

Rewrite the inequality in a more recognizable form. Given inequality is: \[ x^2 - 1 < 0 \] Rewrite it as: \[ x^2 < 1 \]

- Recognize the Structure

Notice that the inequality can be considered as the result of factoring difference of squares: \[ x^2 - 1 = (x - 1)(x + 1) < 0 \]

- Identify Critical Points

Find the points where the product \[ (x - 1)(x + 1) \] equals zero. These points, called critical points, are: \[ x = 1 \text{ and } x = -1 \]

- Test Intervals

Divide the number line into intervals based on the critical points: \[ (-\infty, -1), (-1, 1), (1, \infty) \] Test a point from each interval in the inequality \[ (x - 1)(x + 1) < 0 \] - For interval \[ (-\infty, -1) \]: test point \[ x = -2 \]: \[ (-2 - 1)(-2 + 1) = (-3)(-1) = 3 \], which is not less than 0.- For interval \[ (-1, 1) \]: test point \[ x = 0 \]: \[ (0 - 1)(0 + 1) = (-1)(1) = -1 \], which is less than 0. - For interval \[ (1, \infty) \]: test point \[ x = 2 \]: \[ (2 - 1)(2 + 1) = (1)(3) = 3 \], which is not less than 0.

- Write the Solution Set

Combine the intervals where the product is less than 0. Only the interval \[ (-1, 1) \] satisfies the inequality. So the solution set for the inequality \[ x^2 - 1 < 0 \] is: \[ -1 < x < 1 \]

Key concepts

quadratic inequalities

Quadratic inequalities are expressions involving a quadratic polynomial set within an inequality symbol (<, ≤, >, or ≥). For example, the expression in our problem is \( x^2 - 1 < 0 \). Solving quadratic inequalities involves finding the set of values for the variable that makes the inequality true.

These inequalities are slightly different from quadratic equations, which are solved for exact values where the polynomial equals zero. Here, we are looking for ranges of values. Understanding the differences between inequalities and equations is crucial for solving these types of problems.

Common steps include rewriting the inequality, recognizing its structure, identifying critical points, testing intervals, and finally writing the solution set. Each step builds upon the last to pinpoint the range where the inequality holds true.

factoring

Factoring is a key method used to simplify quadratic inequalities. For quadratic expressions, this often means recognizing and applying the difference of squares. In the exercise, \( x^2 - 1 \) is factored into \( (x - 1)(x + 1) \).

Factoring transforms the quadratic expression from something complex to a product of simpler linear terms. This makes it easier to handle, especially when we later test for intervals. These simpler expressions are key for understanding where the inequality changes.

Recognizing a factorable form such as the difference of squares can save significant time and make solving the inequality straightforward. Always look for factoring opportunities in quadratic inequalities, as they simplify the process immensely.

interval testing

Interval testing is a strategy to determine where the inequality holds true. After factoring, the inequality \( (x - 1)(x + 1) < 0 \) provides us with critical points, which we use to divide the number line into intervals.

The intervals created in our example are \( (-\text{infinity}, -1) \), \((-1, 1)\), and \( (1, \text{infinity}) \). For each of these intervals, we select test points to substitute back into the inequality. The goal is to determine whether the inequality is true or false within each interval.

By testing these points, we find the interval that satisfies the inequality. In our exercise, only the interval \((-1, 1)\) makes \( (x - 1)(x + 1) < 0 \) true. This approach ensures we have considered the entire number line and not overlooked any potential solutions.

critical points

Critical points are values for the variable where the factored expression equals zero. They represent boundaries between intervals where the inequality may change from true to false or vice versa.

For the inequality \( x^2 - 1 < 0 \), the factored form \( (x - 1)(x + 1) \) = 0 gives us critical points at \( x = 1 \) and \( x = -1 \). These points help partition the number line into testable sections.

Critical points are vital because they are often where the inequality transitions between positive and negative values. Identifying and using these points ensures that we have accurately tested the entire range of possible solutions.

In summary, finding and using critical points is a foundational step in solving quadratic inequalities efficiently and accurately.