Problem 30 - Precalculus : Building Concepts and Connections

Question

Solve the inequality algebraically or graphically. $$-x^{2}-3 x>-1$$

Step-by-Step Solution

Answer

The solution to the inequality \(-x^2-3x>-1\) is \((-∞, -1 ) \cup (1, ∞)\).

Step 1: Rearrange the inequality

The first step is to form the inequality into a more familiar quadratic form. Add 1 to both sides to move constant term to one side, resulting in \(-x^2 - 3x + 1 < 0\).

Step 2: Factor the quadratic

Next, factoring the quadratic part will simplify the equation and allow to find its roots. After factoring, the inequality becomes \(-(x - 1)(x + 1) < 0\).

Step 3: Solve the inequality

To solve the inequality, first find the roots by setting the factors equal to zero. This gives \(x = 1\) and \(x = -1\). To find intervals, test the values from each interval in the original inequality. Choose test points less than -1, between -1 and 1, and greater than 1. For example, if -2, 0, and 2 are chosen for \(x\) respectively: \(2 > 0 , -1 < 0 , 2 > 0\).

Step 4: Write the solution

Based on the tests, the solution to the inequality are the values where the inequality holds true. Therefore, the solution intervals, in interval notation, are \((-∞, -1 ) \cup (1, ∞)\).

Key Concepts

Understanding Inequalities

Inequalities represent a relationship between two expressions that are not necessarily equal to each other. In this context, an inequality could be something like \(a < b\) or \(a > b\). When dealing with inequalities involving quadratics, the focus is typically on determining the set of values for which the inequality holds true. It involves finding intervals where the quadratic expression is positive, negative, or zero.

In inequalities, we often rearrange terms to make the problem easier to handle. The primary goal is to express the inequality in a standard form, making it easier to analyze and solve. In this exercise, we rearranged \(-x^{2} - 3x > -1\) into \(-x^{2} - 3x + 1 < 0\) by adjusting the constant term. This was a crucial step to set up the inequality in a way that allows factoring.

Factoring Quadratics

Factoring quadratics is an essential skill, especially when dealing with quadratic inequalities. It involves breaking down a quadratic expression into a product of linear factors, making it easier to identify the roots.

In our problem, we factor \(-x^{2} - 3x + 1\) into \(-(x - 1)(x + 1)\). This means we have transformed the quadratic inequality into a factored form, where each factor represents a linear equation.

• The factored form helps in visualizing how the quadratic function changes signs over different intervals.
• By setting each factor equal to zero, we can find the roots of the equation, or points where the expression equals zero. For this exercise, the roots are \(x = 1\) and \(x = -1\).

This crucial step gives us the points to test various intervals for solving the inequality.

Using Interval Notation

Interval notation is a shorthand way of describing subsets of the real number line. It is especially useful in expressing solutions to inequalities, showing where the inequality holds true.

When the inequality \(-(x - 1)(x + 1) < 0\) was solved, we found that the intervals where the inequality holds were between the roots we found (\(x = -1\) and \(x = 1\)).

• Intervals are often denoted with parentheses \((a, b)\) to reflect that the end points are not included, or brackets \([a, b]\) if the end points are included.
• For open intervals like those in this exercise, we use \((-\infty, -1) \, \cup \, (1, \infty)\), indicating that \(x = -1\) and \(x = 1\) themselves are not included in the solution.

Understanding and accurately writing interval notation is vital for clearly communicating the results of solving quadratic inequalities.