Problem 12 - Algebra 2 and Trigonometry

Question

In \(12-17,\) use a graph to find the solution set of each inequality. $$ x^{2}+2 x-3 < 0 $$

Step-by-Step Solution

Answer

The solution set is \(-3 < x < 1\).

Step 1: Rewrite the Inequality as an Equation

First, treat the inequality as an equation to find the critical points where the graph crosses the x-axis. We'll solve the equation \(x^2 + 2x - 3 = 0\).

Step 2: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 2\), and \(c = -3\). This gives:\[x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-3)}}{2(1)}\] \[x = \frac{-2 \pm \sqrt{4 + 12}}{2}\] \[x = \frac{-2 \pm \sqrt{16}}{2}\] \[x = \frac{-2 \pm 4}{2}\].

Step 3: Solve for x-values from the Quadratic Formula

Calculate the two solutions from the quadratic formula:\[x = \frac{-2 + 4}{2} = 1\] \[x = \frac{-2 - 4}{2} = -3\]. These are the points where the graph intersects the x-axis.

Step 4: Test Intervals Between Critical Points

The x-axis is divided into three intervals based on the roots: 1. \(x < -3\),2. \(-3 < x < 1\),3. \(x > 1\).Test a point in each interval to determine if \(x^2 + 2x - 3 < 0\).

Step 5: Evaluate the Inequality in Different Intervals

Choose a test point from each interval:- For \(x < -3\), choose \(x = -4\): \((-4)^2 + 2(-4) - 3 = 16 - 8 - 3 = 5\) (not less than zero).- For \(-3 < x < 1\), choose \(x = 0\): \(0^2 + 2(0) - 3 = -3\) (which is less than zero).- For \(x > 1\), choose \(x = 2\): \(2^2 + 2(2) - 3 = 4 + 4 - 3 = 5\) (not less than zero).Thus, the solution is the interval where \(x^2 + 2x - 3\) is less than zero, i.e., \(-3 < x < 1\).

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations, typically written as ax^2 + bx + c = 0. It offers a way to find the roots of these equations using the formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, the variables a, b, and c are coefficients from the quadratic equation. By substituting these values into the formula, we calculate the roots where the quadratic equation equals zero.
This method is especially useful when the equation cannot be easily factored.

• Ensure the equation is in the standard form of ax^2 + bx + c = 0 before using the formula.
• The term under the square root, \(b^2 - 4ac\), is called the discriminant. It tells us about the nature of the roots (real or imaginary).
• If the discriminant is positive, there are two distinct real roots. If it is zero, there is one real root. If negative, the roots are complex.

For our original equation, \(x^2 + 2x - 3 = 0\), applying the quadratic formula gives the roots \(x = 1\) and \(x = -3\). These points will help us in further steps to solve the inequality.

Inequality Solutions

In solving inequalities like \(x^2 + 2x - 3 < 0\), the primary goal is to find where the quadratic expression is less than zero. After finding the roots of the related equation with the quadratic formula:

• We obtained the critical points \(x = 1\) and \(x = -3\).

Next, these critical points divide the number line into intervals. Our task is to test each interval to identify where the inequality holds true.
For simple understanding, choose numbers from each interval and substitute them into the inequality:

• If choosing a number less than -3 (say \(x = -4\)), the inequality does not hold as the result is positive.
• Between -3 and 1 (say \(x = 0\)), the inequality holds as it gives a negative value.
• Greater than 1 (say \(x = 2\)), the result again turns positive, therefore not satisfying the inequality.

Thus, the solution to the inequality is the interval \(-3 < x < 1\) where the graph of the quadratic function lies below the x-axis.

Graphical Solution

Visualizing quadratic inequalities graphically can provide insight into where they hold true. When a quadratic equation such as \(x^2 + 2x - 3\) is plotted, it forms a parabola. The nature of parabola, whether it opens upwards or downwards, determines multiple features:

• If the leading coefficient (a) is positive, as in our case, the parabola opens upwards.
• The intersection points with the x-axis, found at critical points \(x = 1\) and \(x = -3\), indicate where the parabola crosses and equals zero.

The problem asks where this function is less than zero, so it is essentially where the parabola is below the x-axis. This occurs between the x-intercepts in the range \(-3 < x < 1\). Additionally, by plotting or analyzing the graph, we can quickly identify intervals that satisfy inequalities without computing values for every point. Understanding this visual approach can greatly enhance your comprehension of quadratic inequalities.

Critical Points

Critical points in quadratic expressions give us significant information about the behavior of the graph. Their primary role is determining where the graph changes direction regarding the x-axis:

• They are derived from setting the quadratic expression equal to zero, which is why we first solve \(x^2 + 2x - 3 = 0\).
• The roots, \(x = 1\) and \(x = -3\), become these critical points.

It's important to understand that critical points mark where the parabola shifts from being above the x-axis to below, or vice versa. This defines enclaves of positive and negative sections of the quadratic plot.
In the context of our inequality problem, critical points help us delineate the intervals for testing values alongside the axis. Identifying these points is a valuable step toward solving inequalities efficiently by enabling formation of testable intervals, like \(-3 < x < 1\) for the inequality \(x^2 + 2x - 3 < 0\). By focusing on critical points, you develop a deeper understanding of how quadratic graphs behave and can predict other important features such as maximum and minimum values if required.