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 questions: 1. Solve the inequality (2)/(3)(x-2) - (1)/(4)(x-5) 0. Step 1: Find the least common multiple (LCM) of the denominators (3 and 4), which is 12. Multiply the entire inequality by 12 to eliminate the fractions. 12 ( (2)/(3)(x-2) - (1)/(4)(x-5) ) 12(0) 8(x-2) - 3(x-5) 0 Step 2: Distribute the numbers into the parentheses. 8x - 16 - 3x + 15 0 Step 3: Combine like terms. (8x - 3x) + (-16 + 15) 0 5x - 1 0 Step 4: Isolate x by adding 1 to both sides and then dividing by 5. 5x 1 x (1)/(5) The solution to the inequality is x (1)/(5). 2. Find the range of x satisfying both inequalities: 2x+2 < 6 and 7x+4 24. Step 1: Solve the first inequality. 2x+2 < 6 2x < 6 - 2 2x < 4 x < 2 Step 2: Solve the second inequality. 7x+4 24 7x 24 - 4 7x 20 x (20)/(7) x 2.857... Step 3: Determine the range of x that satisfies both conditions. We need x < 2 AND x (20)/(7). Since (20)/(7) ≈ 2.86, there is no value of x that can be simultaneously less than 2 and greater than or equal to 2.86. Therefore, there is no range of x that satisfies both inequalities. 3. Find the mean deviation of the numbers 2, 3, 5, 6. Step 1: Calculate the mean (x) of the given numbers. x = (2+3+5+6)/(4) = (16)/(4) = 4 Step 2: Calculate the absolute deviation of each number from the mean (|x - x|). |2 - 4| = |-2| = 2 |3 - 4| = |-1| = 1 |5 - 4| = |1| = 1 |6 - 4| = |2| = 2 Step 3: Sum the absolute deviations. |x - x| = 2 + 1 + 1 + 2 = 6 Step 4: Calculate the mean deviation by dividing the sum of absolute deviations by the number of data points (4). Mean Deviation = |x - x|N = (6)/(4) = 1.5 The mean deviation is 1.5. 4. What is the median and semi-interquartile range of the distribution below? Scores (x) | Frequency (f) ---|--- 1 | 8 2 | 11 3 | 13 4 | 14 5 | 5 6 | 4 7 | 5 8 | 8 Step 1: Calculate the cumulative frequency (cf) and total frequency (N). Scores (x) | Frequency (f) | Cumulative Frequency (cf) ---|---|--- 1 | 8 | 8 2 | 11 | 19 3 | 13 | 32 4 | 14 | 46 5 | 5 | 51 6 | 4 | 55 7 | 5 | 60 8 | 8 | 68 Total frequency (N) = 68. Step 2: Find the median (Q_2). The median is the value at the (N+1)/(2)-th position. Position of Q_2 = (68+1)/(2) = (69)/(2) = 34.5-th position From the cumulative frequency, the 34.5-th value falls within the score of 4 (since cf=32 for score 3 and cf=46 for score 4). So, the median (Q_2) = 4. Step 3: Find the first quartile (Q_1). The first quartile is the value at the (N+1)/(4)-th position. Position of Q_1 = (68+1)/(4) = (69)/(4) = 17.25-th position From the cumulative frequency, the 17.25-th value falls within the score of 2 (since cf=8 for score 1 and cf=19 for score 2). So, Q_1 = 2. Step 4: Find the third quartile (Q_3). The third quartile is the value at the (3(N+1))/(4)-th position. Position of Q_3 = (3(68+1))/(4) = (3 × 69)/(4) = (207)/(4) = 51.75-th position From the cumulative frequency, the 51.75-th value falls within the score of 6 (since cf=51 for score 5 and cf=55 for score 6). So, Q_3 = 6. Step 5: Calculate the semi-interquartile range (SIQR). SIQR = (Q_3 - Q_1)/(2) = (6 - 2)/(2) = (4)/(2) = 2 The median is 4 and the semi-interquartile range is 2. 5. The average age of 10 boys was 12 years. A boy of 14 years was replaced with that of 6 years. Find the new average age of boys. Step 1: Calculate the total initial age of the 10 boys. Total initial age = Average age × Number of boys Total initial age = 12 years × 10 = 120 years Step 2: Calculate the change in total age due to the replacement. Change in total age = Age of new boy - Age of old boy Change in total age = 6 years - 14 years = -8 years Step 3: Calculate the new total age. New total age = Total initial age + Change in total age New total age = 120 years - 8 years = 112 years Step 4: Calculate the new average age. The number of boys remains 10. New average age = New total ageNumber of boys New average age = 112 years10 = 11.2 years The new average age of the boys is 11.2 years. Got more? Send 'em!