Problem 69 - Algebra for College Students

Question

Solve each inequality. State the solution set using interval notation when possible. \(z^{2} \geq 4(z+3)\)

Step-by-Step Solution

Answer

The solution set is \((-∞, -2] union [6, ∞)\).

Step 1: Move All Terms to One Side

Start by subtracting both terms on the right-hand side from the left-hand side to set the inequality to zero: \[z^{2} - 4(z+3) \geq 0\]

Step 3: Factor the Quadratic Expression

Factor the expression. \[z^{2} - 4z - 12 = (z - 6)(z + 2)\]

Step 5: Determine the Intervals

The critical points divide the number line into three intervals. Solve the inequality on each of these intervals: 1. \( (-eq«, -2) \) 2. \( (-2, 6) \) 3. \( (6, eq«) \)

Step 6: Test Each Interval

Choose a test point in each interval and plug it into the inequality to determine if it satisfies the inequality: 1. For \(z = -3\) in \((- eq«, -2) \): \[(-3 - 6)(-3 + 2) > 0 \implies -9 > 0\] 2. For \(z = 0\) in \((-2, 6)\): \[(0 - 6)(0 + 2) < 0 \implies -12 < 0\] 3. For \(z = 7\) in \((6, eq«)\): \[ (7 - 6)(7 + 2) > 0 \implies 9 > 0\]

Step 7: Include Boundaries

Consider the critical points, as they are included in the solution set due to the non-strict inequality (\(\geq\)): \(z = 6\) and \(z = -2\)

Step 8: Write Solution in Interval Notation

Combine all intervals that satisfy the inequality: The solution set in interval notation is: \[(-∞, -2] union [6, ∞)\]

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared with a value (like zero). A quadratic polynomial has the general form:

• ax² + bx + c

We will solve inequalities such as

• ax² + bx + c ≥ 0
• ax² + bx + c ≤ 0

To solve a quadratic inequality, we:

• Rearrange the terms to set the inequality to zero.
• Factor the quadratic expression where possible.
• Determine the intervals by solving the equality ax² + bx + c = 0.
• Test points in each interval to determine where the inequality holds true.

Remember the critical points (solutions of the equation) divide our number line into intervals we must consider. Lastly, include the boundaries when the inequality is non-strict (≥ or ≤).

Interval Notation

Interval notation helps us express a range of values compactly. We use round brackets

• ( and )

for exclusive boundaries and square brackets

• [ and ]

for inclusive boundaries. For example, (-3, 4) means the values between -3 and 4, not including -3 and 4, whereas [-3, 4] means the values between -3 and 4, including both -3 and 4.
Notation (∞) or (-∞) indicates all values greater than or less than a certain point. For instance, (-∞, 5) represents all numbers less than 5. To express a union of two intervals, use the union symbol

• (∪)

For example, [-2, -1) ∪ (3, 4] means the values from -2 to -1 (excluding -1) and (3, 4), including 3 and 4.

Factoring

Factoring helps break down complex quadratic expressions into simpler, solvable forms. For example, factoring

• z² - 4(z + 3)

First, simplify the expression:

• z² - 4z - 12 = 0

Next, identify pairs of factors that multiply to give the constant term (-12) and add up to the coefficient of the linear term (-4). Here, the factors are

• (z - 6)(z + 2)

The roots, or solutions, are where these factors equal zero (z - 6 = 0 and z + 2 = 0). Thus, z = 6 and z = -2 are critical points. These points split the number line into three intervals. Testing each interval helps determine where the inequality holds true. This method simplifies solving quadratic inequalities.