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
Here are the solutions for the given inequalities, including the set solution and number line description: *a) Solve -6 > 2x > 2 Step 1: Split the compound inequality into two separate inequalities. -6 > 2x and 2x > 2 Step 2: Solve each inequality for x. (-6)/(2) > (2x)/(2) -3 > x x < -3 (2x)/(2) > (2)/(2) x > 1 Step 3: Combine the solutions. We need x < -3 and x > 1. There is no value of x that can satisfy both conditions simultaneously. The solution is no solution. No solution Number line representation: An open circle at -3 with shading to the left, and an open circle at 1 with shading to the right. There is no overlap. Set solution: (the empty set) *b) Solve -12 < 3x < 6 Step 1: Divide all parts of the inequality by 3. (-12)/(3) < (3x)/(3) < (6)/(3) Step 2: Simplify the inequality. -4 < x < 2 Number line representation: An open circle at -4 and an open circle at 2, with the region between them shaded. Set solution: (-4, 2) *c) Solve -1 x+2 7 Step 1: Subtract 2 from all parts of the inequality. -1 - 2 x+2 - 2 7 - 2 Step 2: Simplify the inequality. -3 x 5 Number line representation: A closed circle at -3 and a closed circle at 5, with the region between them shaded. Set solution: [-3, 5] *d) Solve -5 < x-1 < 7 Step 1: Add 1 to all parts of the inequality. -5 + 1 < x-1 + 1 < 7 + 1 Step 2: Simplify the inequality. -4 < x < 8 Number line representation: An open circle at -4 and an open circle at 8, with the region between them shaded. Set solution: (-4, 8) *e) Solve -3 3x+3 9 Step 1: Subtract 3 from all parts of the inequality. -3 - 3 3x+3 - 3 9 - 3 -6 3x 6 Step 2: Divide all parts of the inequality by 3. (-6)/(3) (3x)/(3) (6)/(3) Step 3: Simplify the inequality. -2 x 2 Number line representation: A closed circle at -2 and a closed circle at 2, with the region between them shaded. Set solution: [-2, 2] *f) Solve -2 x-1 5 Step 1: Add 1 to all parts of the inequality. -2 + 1 x-1 + 1 5 + 1 Step 2: Simplify the inequality. -1 x 6 Number line representation: A closed circle at -1 and a closed circle at 6, with the region between them shaded. Set solution: [-1, 6] *g) Solve 2y+3 < 11 Step 1: Subtract 3 from both sides of the inequality. 2y+3 - 3 < 11 - 3 2y < 8 Step 2: Divide both sides by 2. (2y)/(2) < (8)/(2) Step 3: Simplify the inequality. y < 4 Number line representation: An open circle at 4, with the region to the left of 4 shaded. Set solution: (-, 4) *h) Solve 2x-5 > 4x+7 Step 1: Subtract 2x from both sides of the inequality. 2x-5 - 2x > 4x+7 - 2x -5 > 2x+7 Step 2: Subtract 7 from both sides of the inequality. -5 - 7 > 2x+7 - 7 -12 > 2x Step 3: Divide both sides by 2. (-12)/(2) > (2x)/(2) Step 4: Simplify the inequality. -6 > x or x < -6 Number line representation: An open circle at -6, with the region to the left of -6 shaded. Set solution: (-, -6) 3 done, 2 left today. You're making progress.