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 simultaneous linear inequalities: 1. 2x - 7 > 9 and 6 - x < 8 First inequality: 2x - 7 > 9 Step 1: Add 7 to both sides. 2x - 7 + 7 > 9 + 7 2x > 16 Step 2: Divide both sides by 2. (2x)/(2) > (16)/(2) x > 8 Second inequality: 6 - x < 8 Step 1: Subtract 6 from both sides. 6 - x - 6 < 8 - 6 -x < 2 Step 2: Multiply both sides by -1 and reverse the inequality sign. -x × (-1) > 2 × (-1) x > -2 Combining the solutions x > 8 and x > -2, the common region is where both conditions are met. If x must be greater than 8, it is automatically greater than -2. The solution is x > 8. 2. 4x - 5 > 15 and 3x + 8 > 44 First inequality: 4x - 5 > 15 Step 1: Add 5 to both sides. 4x - 5 + 5 > 15 + 5 4x > 20 Step 2: Divide both sides by 4. (4x)/(4) > (20)/(4) x > 5 Second inequality: 3x + 8 > 44 Step 1: Subtract 8 from both sides. 3x + 8 - 8 > 44 - 8 3x > 36 Step 2: Divide both sides by 3. (3x)/(3) > (36)/(3) x > 12 Combining the solutions x > 5 and x > 12, the common region is where both conditions are met. If x must be greater than 12, it is automatically greater than 5. The solution is x > 12. 3. 1 - 4x 17 and 2x > 5x + 6 First inequality: 1 - 4x 17 Step 1: Subtract 1 from both sides. 1 - 4x - 1 17 - 1 -4x 16 Step 2: Divide both sides by -4 and reverse the inequality sign. (-4x)/(-4) (16)/(-4) x -4 Second inequality: 2x > 5x + 6 Step 1: Subtract 5x from both sides. 2x - 5x > 5x + 6 - 5x -3x > 6 Step 2: Divide both sides by -3 and reverse the inequality sign. (-3x)/(-3) < (6)/(-3) x < -2 Combining the solutions x -4 and x < -2, the common region is where both conditions are met. If x must be less than or equal to -4, it is automatically less than -2. The solution is x -4. 4. 6x + 4 2x - 8 and 6 - 2x 4 - x First inequality: 6x + 4 2x - 8 Step 1: Subtract 2x from both sides. 6x + 4 - 2x 2x - 8 - 2x 4x + 4 -8 Step 2: Subtract 4 from both sides. 4x + 4 - 4 -8 - 4 4x -12 Step 3: Divide both sides by 4. (4x)/(4) (-12)/(4) x -3 Second inequality: 6 - 2x 4 - x Step 1: Add x to both sides. 6 - 2x + x 4 - x + x 6 - x 4 Step 2: Subtract 6 from both sides. 6 - x - 6 4 - 6 -x -2 Step 3: Multiply both sides by -1 and reverse the inequality sign. -x × (-1) -2 × (-1) x 2 Combining the solutions x -3 and x 2, the common region is where x is greater than or equal to -3 AND less than or equal to 2. The solution is -3 x 2. That's 2 down. 3 left today — send the next one.