X²+4x-21>0

To solve the inequality x² + 4x - 21 > 0, you can use either the factoring method or the quadratic formula.

1. Factoring method: Start by factoring the quadratic equation: (x + 7)(x - 3) > 0 Now, you need to examine the sign of the factors on each interval. Set up a number line and mark the points where the factors change sign (where they equal zero).

   Number line: --------------o------------------------o------------------- -7 3 Test each interval (between the marked points) by picking any value within that interval, and substitute it into the inequality to check the sign: For x < -7, let's choose x = -8: (-8 + 7)(-8 - 3) > 0 (-1)(-11) > 0 11 > 0 - True

   For -7 < x < 3, let's choose x = 0: (0 + 7)(0 - 3) > 0 (7)(-3) > 0 -21 > 0 - False

   For x > 3, let's choose x = 4: (4 + 7)(4 - 3) > 0 (11)(1) > 0 11 > 0 - True

   Based on the sign of the factors, the solution is x < -7 or x > 3.

2. Quadratic formula: Use the quadratic formula to find the roots of the quadratic equation: x = (-b ± √(b² - 4ac)) / 2a

   For the equation x² + 4x - 21 = 0, a = 1, b = 4, and c = -21.

   x = (-4 ± √(4² - 4(1)(-21))) / (2(1)) x = (-4 ± √(16 + 84)) / 2 x = (-4 ± √100) / 2 x = (-4 ± 10) / 2

   This gives two solutions: x1 = (-4 + 10) / 2 = 3 x2 = (-4 - 10) / 2 = -7

   Now, examine the intervals just like in the factoring method: (-∞, -7), (-7, 3), and (3, ∞)

   Test each interval by selecting a value within that interval and substitute it into the inequality to check the sign: For x < -7, let's use x = -8: (-8 + 7)(-8 - 3) > 0 (-1)(-11) > 0 11 > 0 - True

   For -7 < x < 3, let's use x = 0: (0 + 7)(0 - 3) > 0 (7)(-3) > 0 -21 > 0 - False

   For x > 3, let's use x = 4: (4 + 7)(4 - 3) > 0 (11)(1) > 0 11 > 0 - True

   Therefore, the solution is x < -7 or x > 3.