Find the solution set for the inequality: x^2 + 3x - 4 > 0.

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Question: Find the solution set for the inequality: x^2 + 3x - 4 > 0.

Options:

Correct Answer: (-∞, -4) ∪ (1, ∞)

Solution:

Step 1: Factor the quadratic: (x - 1)(x + 4) > 0. Step 2: The solution is outside the roots: (-∞, -4) ∪ (1, ∞).

Find the solution set for the inequality: x^2 + 3x - 4 > 0.

Find the solution set for the inequality: x^2 + 3x - 4 > 0.

Step 1: Start with the inequality x^2 + 3x - 4 > 0.
Step 2: Factor the quadratic expression. We need to find two numbers that multiply to -4 and add to 3. The numbers are 4 and -1.
Step 3: Write the factored form: (x - 1)(x + 4) > 0.
Step 4: Identify the roots of the equation (where it equals 0): x - 1 = 0 gives x = 1, and x + 4 = 0 gives x = -4.
Step 5: Plot the roots on a number line: -4 and 1.
Step 6: Determine the intervals created by the roots: (-∞, -4), (-4, 1), and (1, ∞).
Step 7: Test a point from each interval to see where the inequality holds true: Choose -5 for (-∞, -4), 0 for (-4, 1), and 2 for (1, ∞).
Step 8: Check the signs: For -5, (x - 1)(x + 4) is positive; for 0, it is negative; for 2, it is positive.
Step 9: The solution set is where the product is positive: (-∞, -4) and (1, ∞).

Quadratic Inequalities – Understanding how to solve inequalities involving quadratic expressions by factoring and analyzing sign changes.
Factoring – The ability to factor quadratic expressions to find critical points (roots) for testing intervals.
Interval Notation – Using interval notation to express solution sets for inequalities.