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 to the inequalities, including a description of their representation on a number line: *a) Solve -6 > 2x > 2 Step 1: Split the compound inequality into two separate inequalities and solve for x. -6 > 2x x < -3 2x > 2 x > 1 Step 2: Combine the solutions. We need x < -3 AND x > 1. There is no value of x that satisfies both conditions simultaneously. The solution set is empty. No solution Number line representation: Place an open circle at -3 and shade to the left. Place an open circle at 1 and shade to the right. There is no overlapping region. *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: Place an open circle at -4 and an open circle at 2. Shade the region between -4 and 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: Place a closed circle at -3 and a closed circle at 5. Shade the region between -3 and 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: Place an open circle at -4 and an open circle at 8. Shade the region between -4 and 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: Place a closed circle at -2 and a closed circle at 2. Shade the region between -2 and 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: Place a closed circle at -1 and a closed circle at 6. Shade the region between -1 and 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: Place an open circle at 4. Shade the region to the left of 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: Place an open circle at -6. Shade the region to the left of -6. That's 2 down. 3 left today — send the next one.