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
1.1 Solve the following inequalities and represent your solutions on a number line. 1.1.1 4x - 4 > 0 Step 1: Add 4 to both sides. 4x > 4 Step 2: Divide by 4. x > 1 Number line representation: An open circle at 1, with a line extending to the right. Solution: x > 1 1.1.2 2x + 4 8 Step 1: Subtract 4 from both sides. 2x 4 Step 2: Divide by 2. x 2 Number line representation: A closed circle at 2, with a line extending to the right. Solution: x 2 1.1.3 2x + 6 < 0 Step 1: Subtract 6 from both sides. 2x < -6 Step 2: Divide by 2. x < -3 Number line representation: An open circle at -3, with a line extending to the left. Solution: x < -3 1.1.4 -3x - 8 -5 Step 1: Add 8 to both sides. -3x 3 Step 2: Divide by -3 and reverse the inequality sign. x -1 Number line representation: A closed circle at -1, with a line extending to the right. Solution: x -1 1.2 Solve the following inequalities; represent the solutions of each on a number line and explain the solution of each equation in word. 1.2.1 -3 3x 6 Step 1: Divide all parts by 3. (-3)/(3) (3x)/(3) (6)/(3) -1 x 2 Number line representation: A closed circle at -1 and a closed circle at 2, with a line connecting them. Explanation in words: x is greater than or equal to -1 and less than or equal to 2. Solution: -1 x 2 1.2.2 -1 x - 3 < 5 Step 1: Add 3 to all parts. -1 + 3 x - 3 + 3 < 5 + 3 2 x < 8 Number line representation: A closed circle at 2 and an open circle at 8, with a line connecting them. Explanation in words: x is greater than or equal to 2 and less than 8. Solution: 2 x < 8 1.2.3 1 1 - 2x < 7 Step 1: Subtract 1 from all parts. 1 - 1 1 - 2x - 1 < 7 - 1 0 -2x < 6 Step 2: Divide all parts by -2 and reverse the inequality signs. (0)/(-2) (-2x)/(-2) > (6)/(-2) 0 x > -3 Step 3: Rewrite in standard order (smallest to largest). -3 < x 0 Number line representation: An open circle at -3 and a closed circle at 0, with a line connecting them. Explanation in words: x is greater than -3 and less than or equal to 0. Solution: -3 < x 0 1.3 Represent the following set of linear inequalities on a number line and shade the area where the two inequalities intersect where applicable. Represent the shaded area in set builder notation as in Question 1.2. 1.3.1 x > -2 and x 1 Number line representation: An open circle at -2 and a closed circle at 1, with the line segment between them shaded. Set builder notation: \x -2 < x 1\ 1.3.2 x < -1 and x > 1 Number line representation: An open circle at -1 with a line extending left, and an open circle at 1 with a line extending right. There is no overlap between these two regions. Set builder notation: (empty set, as there is no intersection) 1.3.3 x 6 and x > -1 Number line representation: An open circle at -1 and a closed circle at 6, with the line segment between them shaded. Set builder notation: \x -1 < x 6\ 1.3.4 x -2 and x 3 Number line representation: A closed circle at -2 with a line extending left, and a closed circle at 3 with a line extending right. There is no overlap between these two regions. Set builder notation: (empty set, as there is no intersection)