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
You're on a roll — 1. Solve the inequalities and represent on a number line i) -4x > 12 Step 1: Divide both sides by -4. Remember to reverse the inequality sign when dividing by a negative number. (-4x)/(-4) < (12)/(-4) x < -3 Step 2: Represent on a number line. An open circle at -3 indicates that -3 is not included, and the arrow points to the left. <--------------------o -3 The solution in interval notation is (-, -3). ii) 5x - 7 3x + 9 Step 1: Subtract 3x from both sides. 5x - 3x - 7 9 2x - 7 9 Step 2: Add 7 to both sides. 2x 9 + 7 2x 16 Step 3: Divide both sides by 2. (2x)/(2) (16)/(2) x 8 Step 4: Represent on a number line. A closed circle at 8 indicates that 8 is included, and the arrow points to the right. --------------------> 8 The solution in interval notation is [8, ). iii) 3x - 5 7 Step 1: Add 5 to both sides. 3x 7 + 5 3x 12 Step 2: Divide both sides by 3. (3x)/(3) (12)/(3) x 4 Step 3: Represent on a number line. A closed circle at 4 indicates that 4 is included, and the arrow points to the left. <-------------------- 4 The solution in interval notation is (-, 4]. 2. Solve the compound inequality: -3 2x - 1 < 7 and express your solution set interval notation Step 1: Add 1 to all parts of the inequality. -3 + 1 2x - 1 + 1 < 7 + 1 -2 2x < 8 Step 2: Divide all parts by 2. (-2)/(2) (2x)/(2) < (8)/(2) -1 x < 4 Step 3: Express the solution in interval notation. [-1, 4) 3. A trader buys x bags of rice at N8,000 each and spends an additional N4,000 on transportation. He sells all the bags for N10,000 each. a) Write an inequality to represent the condition that the trader makes a profit. Step 1: Calculate the total cost (Cost Price). Cost Price (CP) = (Cost per bag × number of bags) + Transportation cost CP = 8000x + 4000 Step 2: Calculate the total revenue (Selling Price). Selling Price (SP) = Selling price per bag × number of bags SP = 10000x Step 3: For a profit to be made, the Selling Price must be greater than the Cost Price (SP > CP). 10000x > 8000x + 4000 This is the inequality representing the condition for profit. 10000x > 8000x + 4000 b) Solve the inequality and interpret your answer. Step 1: Subtract 8000x from both sides of the inequality. 10000x - 8000x > 4000 2000x > 4000 Step 2: Divide both sides by 2000. (2000x)/(2000) > (4000)/(2000) x > 2 Step 3: Interpret the answer. The trader makes a profit if he sells more than 2 bags of rice. Since the number of bags must be a whole number, he must sell at least 3 bags to make a profit. x > 2 (The trader must sell more than 2 bags of rice to make a profit.) 4. Solve this inequality: (2x - 1)/(3) (x + 5)/(2) Step 1: Multiply both sides by the least common multiple (LCM) of the denominators 3 and 2, which is 6. 6 ( (2x - 1)/(3) ) 6 ( (x + 5)/(2) ) Step 2: Simplify both sides. 2(2x - 1) 3(x + 5) Step 3: Distribute the numbers on both sides. 4x - 2 3x + 15 Step 4: Subtract 3x from both sides. 4x - 3x - 2 15 x - 2 15 Step 5: Add 2 to both sides. x 15 + 2 x 17 The solution in interval notation is (-, 17]. x 17 Got more? Send 'em!