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, along with a table for symbol meanings and graph representations. 1) Solve 3x^2 + 6 42 - 2x^2 Step 1: Move all terms to one side. 3x^2 + 2x^2 + 6 - 42 0 5x^2 - 36 0 Step 2: Find the critical points by setting the expression equal to zero. 5x^2 - 36 = 0 5x^2 = 36 x^2 = (36)/(5) x = ±sqrt((36)/(5)) x = ±sqrt(36)sqrt(5) x = ±(6)/(sqrt(5)) Step 3: Rationalize the denominator. x = ±6sqrt(5)5 Step 4: Determine the intervals that satisfy the inequality. Since the parabola y = 5x^2 - 36 opens upwards, 5x^2 - 36 0 when x is outside the roots. The solution is x -6sqrt(5)5 or x 6sqrt(5)5. 2) Solve 4x - 2 < 2 Step 1: Add 2 to both sides of the inequality. 4x - 2 + 2 < 2 + 2 4x < 4 Step 2: Divide both sides by 4. (4x)/(4) < (4)/(4) x < 1 The solution is x < 1. 3) Solve 2x - 5 < 5 Step 1: Add 5 to both sides of the inequality. 2x - 5 + 5 < 5 + 5 2x < 10 Step 2: Divide both sides by 2. (2x)/(2) < (10)/(2) x < 5 The solution is x < 5. --- Table of Inequality Symbols and Graph Representation | Symbol | Meaning | Graph Representation (on a number line)