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Section 3.5 Solving Inequalities with Variables on Both Sides California Standards 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12. 5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. Some inequalities have variable terms on both sides of the inequality symbol.You can solve these inequalities like you solved equations with variables on both sides. Use the properties of inequality to “collect” all the variable terms on one side and all the constant terms on the other side. Don’t call me after midnight 1. D= Distributive property 2. C= combine like term 3. M = move variable to one side 4. A = addition/subtraction 5. M = Multiplication/division Solve the Inequality then GRAPH the solution Steps: -Distribute 2x + 8 < 6x -Combine Like Terms -Move Variable 2x + 8 < 6x -2x -2x 8 < 4x 4 -The opposite of adding 2x is subtracting 2x. -Undo multiplication… 4 -…so divide by four 2 < x -4 -2 0 -Graph 2 4 Now You Try One… Solve the Inequality then GRAPH the solution Steps: -Distribute 3x - 12 < 6x -Combine Like Terms -Move Variable 3x - 12 < 6x -3x -3x -12 < 3x 3 -The opposite of adding 3x is subtracting 3x. -Undo multiplication… 3 -…so divide by three -4 < x -4 -2 0 -Graph 2 4 Lets try one more Solve the Inequality then GRAPH the solution 5t + 1 < –2t – 6 Steps: -Distribute 5t + 1 < –2t – 6 +2t +2t 7t + 1 < -6 -1 -1 7t < -7 -Combine Like Terms -Move Variable -The opposite of Negative 2t is Adding 2t. t < -1 –5 –4 –3 –2 –1 0 1 2 Graph 3 4 5 Now You Try Solve and Graph 1. 2x > 4x – 6 2. 7y + 1 < y – 5 3. -3r < 10 – r Match the Following 1. Inequality A. mathematical statement that two expressions are equivalent 2. Equation B. value of a variable that makes a statement true 3. Inverse Operations C. terms that contain the same variable raised to the same power 4. Like Terms D. 5. Solution of an Equation a mathematical statement that compares two expressions by using one of the following signs: <, >, <, >, or ≠ E. operation that “undo” each other Let’s try this one together Solve the Inequality then GRAPH the solution Steps: -Distribute 6x < 4(x + 1) 6x < 4x + 4 -Combine Like Terms -Move Variable 6x <4x +4 -4x -4x 2x < 4 2 2 -The opposite of adding 4x is Subtracting 4x. -Undo multiplication… -…so divide by two x < 2 -4 -2 0 -Graph 2 4 Now You Try Solve the Inequality then GRAPH the solution Steps: -Distribute 2(6 – x) < 4x 12 - 2x < 4x -Combine Like Terms -Move Variable 12 – 2x <4x +2x +2x 12 < 6x 6 6 -The opposite of subtracting 2x is adding 2x. -Undo multiplication… -…so divide by six 2 < x -4 -2 0 -Graph 2 4 Lets try one more Solve the Inequality then GRAPH the solution x+5>x+3 x+5>x+3 -x -x 0 +5>0+3 -5 -5 0 -2 0 > -2 All real numbers Steps: -Distribute -Combine Like Terms -Move Variable -The opposite of adding x is subtracting x. -Does it make a true statement? Lets try another one Solve the Inequality then GRAPH the solution 2x + 6 < 5 + 2x Steps: -Distribute 2x + 6 < 5 + 2x -2x -2x 0 +6<5+0 -5 -5 0 0 -Combine Like Terms 1 < 0 -Does it make a true statement? No Solutions -Move Variable -The opposite of adding 2x is subtracting 2x. Now You Try Solve and Graph 1. 4x > 3(7 – x) 2. 2(x – 2) < -2(1 – x) 3. 4(y + 1) < 4y + 2 Lesson Quiz Solve each inequality and graph the solutions. 1. t < 5t + 24 t > –6 2. 5x – 9 ≤ 4.1x – 81 x ≤ –80 3. 4b + 4(1 – b) > b – 9 b < 13 Solve each inequality. 5. 2y – 2 ≥ 2(y + 7) ø 6. 2(–6r – 5) < –3(4r + 2) all real numbers