Problem 103 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. $$(a) x^{2}+4 x+3 \geq 0$$ $$(b) x^{2}+4 x+3<0$$

Step-by-Step Solution

Answer

(a) Solution: \((-\infty, -3] \cup [-1, \infty)\); (b) Solution: \((-3, -1)\).

Step 1: Identify the inequality and rewrite

The inequality given is \(x^2 + 4x + 3 \geq 0\). We will start by rewriting it in a more manageable format: \((x+1)(x+3) \geq 0\).

Step 2: Find critical points

Set each factor in \((x+1)(x+3) = 0\) to find critical points: \(x+1=0\) gives \(x=-1\) and \(x+3=0\) gives \(x=-3\). These are the points where the inequality can change its sign.

Step 3: Test intervals between critical points

The critical points divide the number line into intervals: \((-\infty, -3)\), \((-3, -1)\), and \((-1, \infty)\). Choose test points from each interval, say \(x=-4, x=-2, x=0\), and check if the inequality holds.- For \((-\infty, -3)\), testing \(x=-4\) gives \((x+1)(x+3) = (-4+1)(-4+3) = 3(-1) < 0\).- For \((-3, -1)\), testing \(x=-2\) gives \((x+1)(x+3) = (-2+1)(-2+3) = (-1)(1) < 0\).- For \((-1, \infty)\), testing \(x=0\) gives \((x+1)(x+3) = (0+1)(0+3) = 1 \cdot 3 > 0\).

Step 4: Determine solution set for \(x^2+4x+3 \geq 0\)

The inequality holds for the intervals \((-\infty, -3] \cup [-1, \infty)\) because it changes sign at each critical point and includes the region where the expression is positive or zero.

Step 5: Analyze the inequality \(x^2+4x+3<0\)

Rewrite the inequality as \((x+1)(x+3) < 0\). Using the same critical points \(x=-3\) and \(x=-1\), we analyze the intervals. The inequality holds for \((-3, -1)\), where it evaluates to negative. Hence, the solution is \(-3 < x < -1\).

Step 6: Graphical representation

Graph the parabola \(y = x^2 + 4x + 3\). It has roots at \(x=-3\) and \(x=-1\). The solution to \(x^2+4x+3 \geq 0\) includes the regions where the parabola is on or above the x-axis, and \(x^2+4x+3<0\) is where it's below the x-axis.

Key Concepts

Quadratic Inequality

Understanding quadratic inequalities is crucial in algebra. These inequalities involve polynomial expressions of degree two, typically having the form \(ax^2 + bx + c \geq 0\) or \(ax^2 + bx + c < 0\). To tackle these problems, we need to determine the set of \(x\) values that satisfy the inequality.

Quadratic inequalities can be more challenging than linear ones because they involve parabolic curves. The curves are defined by three regions: where the curve is above, below, or exactly on the x-axis. Rewriting the quadratic in factor form, such as \((x + 1)(x + 3) \geq 0\), simplifies the process of identifying the critical points, which are essential in solving these inequalities.

Critical Points

Critical points in the context of quadratic inequalities are the x-values where the inequality changes its behavior, like toggling between positive and negative. For the quadratic \(x^2 + 4x + 3 = (x+1)(x+3)\), we find these by setting each factor to zero.

Here, solving \((x+1) = 0\) gives \(x = -1\), and \((x+3) = 0\) gives \(x = -3\). These values are the roots of the quadratic equation. At these critical points, the quadratic expression equals zero. Critical points divide the number line into segments or intervals that we need to evaluate individually to solve the inequality. A change in interval from positive to negative helps us determine solutions for \(x\) values that satisfy the inequality.

Graphical Representation

Graphical representation is a way to visually understand quadratic inequalities by plotting the quadratic function as a parabola. For instance, the function \(y = x^2 + 4x + 3\) depicts a parabola with roots at \(x = -3\) and \(x = -1\).

This curve helps to see the portions where it lies above or below the x-axis. If you need to solve \(x^2 + 4x + 3 \geq 0\), you will look for regions where the parabola touches or is above the x-axis. Conversely, for \(x^2 + 4x + 3 < 0\), the focus is where the curve is entirely below the x-axis.
Critical points noted on the graph illustrate these intersections with the x-axis. Thus, it becomes easier to see which intervals meet the inequality’s conditions visually.

Interval Testing

Interval testing is the process of choosing test values from different intervals between critical points to determine where the inequality holds true. After identifying critical points like \(-3\) and \(-1\), the number line is divided into intervals: \((-\infty, -3)\), \((-3, -1)\), and \((-1, \infty)\).

From each interval, pick test points; for example, \(x = -4\), \(x = -2\), and \(x = 0\). Plug these into the factored inequality \((x+1)(x+3)\) and check their signs:

• For \((-\infty, -3)\), \(x = -4\) results in a negative product, so the inequality doesn’t hold.
• For \((-3, -1)\), \(x = -2\) also results in a negative product (meaning \(x^2 + 4x + 3 < 0\) holds).
• For \((-1, \infty)\), \(x = 0\) gives a positive product, meaning \(x^2 + 4x + 3 \geq 0\) is true.

This analysis helps us compile the solutions that fit the original inequality, giving us the correct intervals for \(x\).