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Section 3.5

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(2 x – 5) + 4( x – 2) = 12 .

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Section 3.5

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  1. Section 3.5 Solving Inequalities with Variables on Both Sides

  2. 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.

  3. 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.

  4. 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

  5. 2x + 8 < 6x 2x + 8 < 6x -2x 8 < 4x Steps: -Distribute -Combine Like Terms -Move Variable -The opposite of adding 2x is subtracting 2x. -Undo multiplication… -…so divide by four -Graph Solve the Inequality then GRAPH the solution -2x 4 4 2 < x -4 -2 0 2 4

  6. 3x - 12 < 6x 3x - 12 < 6x -3x -12 < 3x Steps: -Distribute -Combine Like Terms -Move Variable -The opposite of adding 3x is subtracting 3x. -Undo multiplication… -…so divide by three -Graph Now You Try One…Solve the Inequality then GRAPH the solution -3x 3 3 -4 < x -4 -2 0 2 4

  7. 5t + 1 < –2t – 6 5t + 1 < –2t – 6 +2t +2t 7t + 1 < - 6 -1 -1 Steps: -Distribute -Combine Like Terms -Move Variable -The opposite of Negative 2t is Adding 2t. Graph –4 –1 5 –3 –2 0 1 2 3 4 –5 Lets try one moreSolve the Inequality then GRAPH the solution 7t < -7 t < -1

  8. Now You TrySolve and Graph • 2x > 4x – 6 2. 7y + 1 < y – 5 3. -3r < 10 – r

  9. Match the Following • Inequality • Equation • Inverse Operations • Like Terms • Solution of an Equation • mathematical statement that two expressions are equivalent • value of a variable that makes a statement true • terms that contain the same variable raised to the same power • a mathematical statement that compares two expressions by using one of the following signs: <, >, <, >, or ≠ • operation that “undo” each other

  10. 6x < 4(x + 1) 6x < 4x + 4 6x <4x +4 -4x 2x < 4 Steps: -Distribute -Combine Like Terms -Move Variable -The opposite of adding 4x is Subtracting 4x. -Undo multiplication… -…so divide by two -Graph Let’s try this one togetherSolve the Inequality then GRAPH the solution -4x 2 2 x < 2 -4 -2 0 2 4

  11. 2(6 – x) < 4x 12 - 2x < 4x 12 – 2x <4x +2x 12 < 6x Steps: -Distribute -Combine Like Terms -Move Variable -The opposite of subtracting 2x is adding 2x. -Undo multiplication… -…so divide by six -Graph Now You TrySolve the Inequality then GRAPH the solution +2x 6 6 2 < x -4 -2 0 2 4

  12. x + 5 > x + 3 x + 5 > x + 3 -x -x 0 + 5 > 0 + 3 -5 -5 Steps: -Distribute -Combine Like Terms -Move Variable -The opposite of adding x is subtracting x. -Does it make a true statement? Lets try one moreSolve the Inequality then GRAPH the solution 0 -2 0 > -2 All real numbers

  13. 2x + 6 < 5 + 2x 2x + 6 < 5 + 2x -2x -2x 0 + 6 < 5 + 0 -5 -5 Steps: -Distribute -Combine Like Terms -Move Variable -The opposite of adding 2x is subtracting 2x. -Does it make a true statement? Lets try another one Solve the Inequality then GRAPH the solution 0 0 1 < 0 No Solutions

  14. Now You TrySolve and Graph 1. 4x > 3(7 – x) 2. 2(x – 2) < -2(1 – x) 3. 4(y + 1) < 4y + 2

  15. 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

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