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Determine the solution set for the inequality: x^2 + 4x + 3 < 0.

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Question: Determine the solution set for the inequality: x^2 + 4x + 3 < 0.

Options:

  1. (-3, -1)
  2. (-1, 3)
  3. (-∞, -3)
  4. (-∞, -1)

Correct Answer: (-3, -1)

Solution: 

Step 1: Factor the quadratic: (x + 3)(x + 1) < 0. Step 2: The critical points are x = -3 and x = -1. Step 3: Test intervals: The solution set is (-3, -1).

Determine the solution set for the inequality: x^2 + 4x + 3 < 0.

Practice Questions

Q1
Determine the solution set for the inequality: x^2 + 4x + 3 < 0.
  1. (-3, -1)
  2. (-1, 3)
  3. (-∞, -3)
  4. (-∞, -1)

Questions & Step-by-Step Solutions

Determine the solution set for the inequality: x^2 + 4x + 3 < 0.
  • Step 1: Factor the quadratic expression x^2 + 4x + 3. This means we need to find two numbers that multiply to 3 (the constant term) and add to 4 (the coefficient of x). The factors are (x + 3)(x + 1).
  • Step 2: Identify the critical points. These are the values of x that make the expression equal to 0. Set (x + 3) = 0 and (x + 1) = 0. This gives us the critical points x = -3 and x = -1.
  • Step 3: Test the intervals created by the critical points. The intervals are (-∞, -3), (-3, -1), and (-1, ∞). Choose a test point from each interval and substitute it into the factored inequality (x + 3)(x + 1) < 0 to see if the inequality holds true.
  • Step 4: For the interval (-∞, -3), choose x = -4: (-4 + 3)(-4 + 1) = (-1)(-3) = 3 (not less than 0).
  • Step 5: For the interval (-3, -1), choose x = -2: (-2 + 3)(-2 + 1) = (1)(-1) = -1 (less than 0).
  • Step 6: For the interval (-1, ∞), choose x = 0: (0 + 3)(0 + 1) = (3)(1) = 3 (not less than 0).
  • Step 7: The solution set is where the inequality is true, which is the interval (-3, -1).
  • Quadratic Inequalities – Understanding how to solve inequalities involving quadratic expressions by factoring and testing intervals.
  • Critical Points – Identifying points where the expression equals zero to determine intervals for testing the inequality.
  • Interval Testing – Using test points in the intervals defined by critical points to determine where the inequality holds true.
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