Use interval notation to indicate the set of solutions to the inequality x2 - 5x - 6 < 0.

Mathematics
Use interval notation to indicate the set of solutions to the inequality x2 - 5x - 6 < 0.

ScanSolve AI Answer

Answer

[-1,6]

Step 1: Solve the corresponding equation x2−5x−6=0x^2 - 5x - 6 = 0 to find the boundary points.

Apply the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a=1a = 1, b=−5b = -5, c=−6c = -6.

Compute the discriminant:

b2−4ac=(−5)2−4(1)(−6)=25+24=49b^2 - 4ac = (-5)^2 - 4(1)(-6) = 25 + 24 = 49

Substitute:

x=5±492=5±72x = \frac{5 \pm \sqrt{49}}{2} = \frac{5 \pm 7}{2} x=5+72=122=6,x=5−72=−22=−1x = \frac{5 + 7}{2} = \frac{12}{2} = 6, \quad x = \frac{5 - 7}{2} = \frac{-2}{2} = -1

The roots are x=−1x = -1 and x=6x = 6.

Step 2: Factor the quadratic:

x2−5x−6=(x+1)(x−6)x^2 - 5x - 6 = (x + 1)(x - 6)

Step 3: Determine the sign of (x+1)(x−6)(x + 1)(x - 6) in the test intervals (−∞,−1)(-\infty, -1), (−1,6)(-1, 6), and (6,∞)(6, \infty).

  • For x=−2x = -2 in (−∞,−1)(-\infty, -1): (−2+1)(−2−6)=(−1)(−8)=8>0(-2 + 1)(-2 - 6) = (-1)(-8) = 8 > 0
  • For x=0x = 0 in (−1,6)(-1, 6): (0+1)(0−6)=(1)(−6)=−6<0(0 + 1)(0 - 6) = (1)(-6) = -6 < 0
  • For x=7x = 7 in (6,∞)(6, \infty): (7+1)(7−6)=(8)(1)=8>0(7 + 1)(7 - 6) = (8)(1) = 8 > 0

The expression is negative or zero between the roots, and the parabola opens upwards (since a>0a > 0).

Thus, x2−5x−6≤0x^2 - 5x - 6 \leq 0 for −1≤x≤6-1 \leq x \leq 6.

Step 4: Express in interval notation: [−1,6][-1, 6].

Final answer: [-1,6]

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Quick Answer

Solve the corresponding equation x^2 - 5x - 6 = 0 to find the boundary points. Apply the quadratic formula: x = -b ± sqrt(b^2 - 4ac)2a where a = 1, b = -5, c = -6.

Use interval notation to indicate the set of solutions to the inequality x2 - 5x - 6 < 0.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Solve the corresponding equation x^2 - 5x - 6 = 0 to find the boundary points. Apply the quadratic formula: x = -b ± sqrt(b^2 - 4ac)2a where a = 1, b = -5, c = -6. Compute the discriminant: b^2 - 4ac = (-5)^2 - 4(1)(-6) = 25 + 24 = 49 Substitute: x = 5 ± sqrt(49)2 = (5 ± 7)/(2) x = (5 + 7)/(2) = (12)/(2) = 6, x = (5 - 7)/(2) = (-2)/(2) = -1 The roots are x = -1 and x = 6. Step 2: Factor the quadratic: x^2 - 5x - 6 = (x + 1)(x - 6) Step 3: Determine the sign of (x + 1)(x - 6) in the test intervals (-, -1), (-1, 6), and (6, ). For x = -2 in (-, -1): (-2 + 1)(-2 - 6) = (-1)(-8) = 8 > 0 For x = 0 in (-1, 6): (0 + 1)(0 - 6) = (1)(-6) = -6 < 0 For x = 7 in (6, ): (7 + 1)(7 - 6) = (8)(1) = 8 > 0 The expression is negative or zero between the roots, and the parabola opens upwards (since a > 0). Thus, x^2 - 5x - 6 ≤ 0 for -1 ≤ x ≤ 6. Step 4: Express in interval notation: [-1, 6]. Final answer: [-1,6]