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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Here are the solutions to the mathematics problems: SUBJECT: MATHEMATICS 1. a) Simplify (7x - 4) - (4x - 12) Step 1: Distribute the negative sign to the terms in the second parenthesis. (7x - 4) - (4x - 12) = 7x - 4 - 4x + 12 Step 2: Group like terms. (7x - 4x) + (-4 + 12) Step 3: Combine like terms. 3x + 8 The simplified expression is: 3x + 8 b) Solve the equation (x)/(3) + (x)/(2) = (5)/(2) Step 1: Find the least common multiple (LCM) of the denominators (3, 2, 2), which is 6. Step 2: Multiply every term in the equation by the LCM to eliminate the denominators. 6 ((x)/(3)) + 6 ((x)/(2)) = 6 ((5)/(2)) Step 3: Simplify the terms. 2x + 3x = 15 Step 4: Combine like terms on the left side. 5x = 15 Step 5: Divide by 5 to solve for x. x = (15)/(5) x = 3 The solution for x is: x = 3 2. a) Simplify ((x-3)(x+1))/(x) Step 1: Expand the numerator using the FOIL method (First, Outer, Inner, Last). (x-3)(x+1) = x(x) + x(1) - 3(x) - 3(1) = x^2 + x - 3x - 3 = x^2 - 2x - 3 Step 2: Write the expression with the expanded numerator. (x^2 - 2x - 3)/(x) Step 3: Divide each term in the numerator by x. (x^2)/(x) - (2x)/(x) - (3)/(x) x - 2 - (3)/(x) The simplified expression is: x - 2 - (3)/(x) b) Simplify 8x^2 × (3xy)/(6xy^2) Step 1: Multiply the terms. (8x^2 · 3xy)/(6xy^2) Step 2: Multiply the coefficients and combine the variables in the numerator. (24x^3y)/(6xy^2) Step 3: Simplify the coefficients and variables. Divide 24 by 6, x^3 by x, and y by y^2. (24)/(6) · (x^3)/(x) · (y)/(y^2) 4 · x^(3-1) · y^(1-2) 4x^2y^-1 (4x^2)/(y) The simplified expression is: (4x^2)/(y) 3. a) Expand (x+1)(x-2) Step 1: Use the FOIL method (First, Outer, Inner, Last). (x+1)(x-2) = x(x) + x(-2) + 1(x) + 1(-2) = x^2 - 2x + x - 2 Step 2: Combine like terms. = x^2 - x - 2 The expanded expression is: x^2 - x - 2 b) Find the HCF of 4x^2y and 6xyz Step 1: Find the prime factorization of each term. 4x^2y = 2^2 · x^2 · y 6xyz = 2 · 3 · x · y · z Step 2: Identify the common factors and their lowest powers. • Common numerical factor: 2^1 (from 2^2 and 2^1) • Common variable x: x^1 (from x^2 and x^1) • Common variable y: y^1 (from y^1 and y^1) • Variable z is not common. Step 3: Multiply the common factors. HCF = 2 · x · y HCF = 2xy The HCF is: 2xy 4. a) Which of the following inequalities is sketched in the figure below? The figure shows a number line with a closed circle at -1 and the line shaded to the right of -1. • A closed circle indicates that the number -1 is included in the solution set. This means "greater than or equal to" () or "less than or equal to" (). • Shading to the right indicates values greater than -1. Combining these, the inequality represents all numbers x that are greater than or equal to -1. x -1 The inequality is: x -1 That's 2 down. 3 left today — send the next one.