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.
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You're on a roll — 1) Step 1: Set up the equations based on the given information. Let the two numbers be x and y. "The sum of two numbers exceeds their product by one": x + y = xy + 1 (1) "Their difference is equal to their product less five": x - y = xy - 5 (2) Step 2: Solve the system of equations. Subtract equation (2) from equation (1): (x + y) - (x - y) = (xy + 1) - (xy - 5) x + y - x + y = xy + 1 - xy + 5 2y = 6 y = 3 Step 3: Substitute the value of y into equation (1) to find x. x + 3 = x(3) + 1 x + 3 = 3x + 1 3 - 1 = 3x - x 2 = 2x x = 1 Step 4: State the two numbers. The two numbers are 1 and 3. 1 and 3 2) Step 1: Solve the first inequality. 3x - 1 > -4 Add 1 to both sides: 3x > -4 + 1 3x > -3 Divide by 3: x > -1 Step 2: Solve the second inequality. The problem states 2x + 1 7. Subtract 1 from both sides: 2x 7 - 1 2x 6 Divide by 2: x 3 Step 3: Combine the solutions. The values of x must satisfy both x > -1 and x 3. This can be written as a compound inequality: -1 < x 3 3) Step 1: Evaluate the numerator. -12 ÷ (-3) × 4 - (-15) Perform division first: 4 × 4 - (-15) Perform multiplication: 16 - (-15) Simplify subtraction of a negative: 16 + 15 = 31 Step 2: Evaluate the denominator. -5 × 6 ÷ 2 + (-5) Perform multiplication first: -30 ÷ 2 + (-5) Perform division: -15 + (-5) Perform addition: -15 - 5 = -20 Step 3: Form the fraction and simplify. (31)/(-20) = -(31)/(20) -(31)/(20) 4) Step 1: Evaluate the numerator. (-4)(-2) + (-12) - (+3) + -20 + (+4) + -6 Perform multiplication: 8 + (-12) - 3 - 20 + 4 - 6 Combine terms: 8 - 12 - 3 - 20 + 4 - 6 (8 + 4) - (12 + 3 + 20 + 6) 12 - 41 -29 Step 2: Evaluate the denominator. The denominator is given as -9 - (15) 46 - (8+2) - 3. This implies multiplication between the two parts. First part: -9 - 15 = -24 Second part: 46 - (8+2) - 3 46 - 10 - 3 36 - 3 = 33 Multiply the two parts: (-24) × (33) = -792 Step 3: Form the fraction and simplify. (-29)/(-792) = (29)/(792) (29)/(792) 5) Step 1: Evaluate the numerator. -8 - 2 + 12 × 9 - 4 × 6 Perform multiplications: -8 - 2 + 108 - 24 Perform additions and subtractions from left to right: -10 + 108 - 24 98 - 24 74 Step 2: Evaluate the denominator. 56 - 7 × 2 Perform multiplication: 56 - 14 Perform subtraction: 42 Step 3: Form the fraction and simplify. (74)/(42) Divide both numerator and denominator by their greatest common divisor, which is 2: (74 ÷ 2)/(42 ÷ 2) = (37)/(21) (37)/(21) What's next?