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1-5 Solving Inequalities Solving inequalities by addition, subtraction, multiplication, and division Remember: Inequality Signs < -- Less than > -- Greater than ≤ -- Less than or Equal to ≥ -- Greater than or Equal to Graph an Inequality in One Variable x <2 z ≤1 a > -2 0 ≤d Remember Open circle represents everything up to that value but does not include it. Closed circle represents everything up to that value and that value itself. Use open circle for < and >. Use closed circle for ≤ and ≥. Ok… now you try Linear Equations v. Linear Inequalities • • There are many characteristics between inequalities and equations that are very similar… Then again there are differences also… Linear Equations v. Linear Inequalities Equations have equal signs Equations generally have only one solution. We undo operations in order to solve equations. Inequalities use inequality signs. Inequalities represent infinite solutions We undo operations in order to solve inequalities. Solving Equations is very similar to solving Inequalities… 𝑥 + 4 = 7 𝑥+ 4– 4 = 7 − 4 𝑥 = 3 𝑥 + 4 ≥ 7 𝑥 + 4– 4 ≥ 7 − 4 𝑥 ≥ 3 Solving Equations is very similar to solving Inequalities… −2 = 𝑛 − 4 −2 + 4 = 𝑛 – 4 + 4 2 = 𝑛 𝑛 = 2 −2 > 𝑛 − 4 −2 + 4 > 𝑛 – 4 + 4 2 > 𝑛 𝑛 < 2 Set Builder Notation The Solution n < 2 states that set of all numbers less then 2 are solutions to the Inequality in the Example. Another way to represent n < 2 is in set builder notation. {n | n < 2} You try… 1. d+4≤6 𝑑 𝑑 ≤ 2} 4. -2≥h+6 ℎ ℎ ≤ −8} 2. x–3>2 5. - 5 ≤ -5 + s 3. 𝑥 𝑥 > 5} q + 12 ≥ 4 6. 𝑠 𝑠 ≥ 0} 2v > v - 3 𝑞 𝑞 ≥ −8} 𝑣 𝑣 > −3} 6.2 Investigating Inequalities Developing Concepts • How do operations affect an Inequality Create your own! Create any inequality you want that is TRUE! Use either < or >! Fill in the table with your inequality… Think about it… Would I flip the inequality sign if… 6.2 Solving Inequalities by using Multiplication and Division • To solve one step inequalities in one variable using multiplication or division. Look at the symbols! Properties of Inequalities with Multiplication/Division… When Multiplying or Dividing both sides by a positive number (𝑛 > 0)… Keep the Inequality sign the way it is. When multiplying or dividing both sides by a negative number (𝑛 < 0)… Flip the Inequality sign… Properties of Inequalities with Division… When dividing both sides by a positive number… Keep the Inequality sign the way it is. When dividing both sides by a negative number… Flip the Inequality sign… Look at the symbols! Solve 𝑥 < −2 4 𝑥 (4) < −2(4) 4 Multiplied by Positive 𝑥 < −8 𝑥 <6 −3 𝑥 (−3) < −2(−3) −3 Multiplied by Negative 𝑥>6 Solve… −20 ≤ 4𝑥 −20 4𝑥 ≤ 4 4 −5 ≤ 𝑥 𝑥 ≥ −5 20 ≤ −4𝑥 20 −4𝑥 ≤ −4 −4 −5 ≥ 𝑥 𝑥 ≤ −5 You Try… ≤ -4 -5 ≤ ≤3 < -10 5x ≤ -15 -30 < -6x -3x <9 9x > -3 What about Fractions that are Coefficients? 1 x > -6 2 2 1 2 ( ) x > -6( ) 1 2 1 x > -12 2 3 2 3 - x > 18 (- 3 )2 x > 18(- X < -27 3 ) 2 Try this… 6.3 Solving Multi-Step Inequalities Goal: Solving multistep Inequalities in one variable What is a Multi-Step Inequality? A Multi-Step Inequality is just like a Multi-Step Equation. It takes more than one step to solve. Solve Inequalities/Equations Solve… 2y – 5 < 7 2y – 5 + 5 < 7 + 5 2y < 12 2𝑦 12 < 2 2 y<6 Think… 2y – 5 = 7 2y – 5 + 5 = 7 + 5 2y = 12 2𝑦 12 = 2 2 y=6 Solve… Solve… -5 – x > 4 -5 – x + 5 > 4 + 5 -x > 9 (-1) –x > 9 (-1) x < -9 Think… -5 – x = 4 -5 – x + 5 = 4 + 5 -x = 9 (-1) –x = 9 (-1) x = -9 You try… 1. 3x – 5 > 4 3. -4 < y or y > -4 x>3 2. 10 – n ≤ 5 n≥5 13 > - 3 - 4y 4. 𝑥 4 +6≥5 x ≥ -4 Using the Distributive Property… Solve… 3(x + 2) < 7 3x + 6 < 7 3x + 6 - 6 < 7 – 6 3x < 1 < x < Try… 3(n – 4) ≥ 6 -2(x + 1) < 2 What if there were two variables? 2x – 3 ≥ 4x + 1 2x – 3 – 2x ≥ 4x + 1 - 2x -3 ≥ 2x + 1 -3 - 1 ≥ 2x + 1 – 1 -4 ≥ 2x −4 2𝑥 ≥ 2 -2 ≥ x x ≤ -2 2 You try… 5n X – 21 < 8n + 6 + 3 ≥ 2x - 4 4y - – 3 < -y + 12 3z + 15 > 2z Solve the Equations! 4x+8 = 4(x+2) 4x + 8 = 4x + 8 4x + 8 – 4x = 4x + 8 – 4x 8=8 Infinite Solutions Or Identity 2x+10 = -2(-x + 5) 2x+10 = 2x - 10 2x + 10 - 2x = 2x - 10 - 2x 10 = -10 10 ≠ -10 No Solutions Unbalanced! What about if the variables eliminate each other? Solve 4x+8 > 4(x+2) 4x + 8 > 4x + 8 4x + 8 – 4x > 4x + 8 – 4x 8>8 N/S! This inequality will not work because my end statement is NOT TRUE! What if the sign was ≥ ? Lets Try… 2x+6 ≤ 2(x-3) 2x+6 ≤ 2x-6 2x+6 – 2x ≤ 2x-6 – 2x 6 ≤ -6 This is an untrue inequality! Therefore there is No Solutions to this Inequality! Now Try… 2(x + > x - 4 x+>x–4 x+-x>x–4-x >–4 This is a true inequality! Therefore any solution for x will work. It has Infinite Solutions. 1.) 2𝑥 + 5 − 6𝑥 > −4(𝑥 + 5) 5 > -20 TRUE! Infinite Solutions! 2. ) 4 𝑥 − 2 ≥ −2 (4 − 2𝑥) -8 3.) ≥ -8 TRUE! Infinite Solutions! 2x + 4 > 5x – 2 2 > x or x < 2 4.) 5( 1 – x ) ≤ -1( 5x + 10) 5 ≤ - 10 FALSE! NO SOLUTIONS