To solve this inequality, we first factor the denominator:
x^2 + 5x - 14 = (x + 7)(x - 2)
Therefore, the inequality becomes:
(3x - 15) / (x + 7)(x - 2) ≥ 0
Next, we find the critical points by setting the numerator and denominator equal to zero:
3x - 15 = 0x = 5
x + 7 = 0x = -7
x - 2 = 0x = 2
So the critical points are x = -7, 2, and 5.
We can create a number line with these critical points and test intervals between them to determine where the inequality holds true.
• Test x = -8: (-3(-8) - 15) / (-8 + 7)(-8 - 2) = 39 / 60 > 0• Test x = -6: (-3(-6) - 15) / (-6 + 7)(-6 - 2) = 3 / 16 > 0• Test x = 0: (-3(0) - 15) / (0 + 7)(0 - 2) = -15 / -14 > 0• Test x = 3: (-3(3) - 15) / (3 + 7)(3 - 2) = -24 / 30 < 0• Test x = 6: (-3(6) - 15) / (6 + 7)(6 - 2) = -33 / 65 < 0
So, the solution to the inequality is x < -7 or 2 < x ≤ 5 or x > 5.
To solve this inequality, we first factor the denominator:
x^2 + 5x - 14 = (x + 7)(x - 2)
Therefore, the inequality becomes:
(3x - 15) / (x + 7)(x - 2) ≥ 0
Next, we find the critical points by setting the numerator and denominator equal to zero:
3x - 15 = 0
x = 5
x + 7 = 0
x = -7
x - 2 = 0
x = 2
So the critical points are x = -7, 2, and 5.
We can create a number line with these critical points and test intervals between them to determine where the inequality holds true.
• Test x = -8: (-3(-8) - 15) / (-8 + 7)(-8 - 2) = 39 / 60 > 0
• Test x = -6: (-3(-6) - 15) / (-6 + 7)(-6 - 2) = 3 / 16 > 0
• Test x = 0: (-3(0) - 15) / (0 + 7)(0 - 2) = -15 / -14 > 0
• Test x = 3: (-3(3) - 15) / (3 + 7)(3 - 2) = -24 / 30 < 0
• Test x = 6: (-3(6) - 15) / (6 + 7)(6 - 2) = -33 / 65 < 0
So, the solution to the inequality is x < -7 or 2 < x ≤ 5 or x > 5.