 # 3.1-3.2 Solving Inequalities Using Addition & Subtraction.

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3.1-3.2 Solving Inequalities Using Addition & Subtraction

An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than ≤ : less than or equal to > : greater than ≥ : greater than or equal to

“x < 5” means that whatever value x has, it must be less than 5. Try to name ten numbers that are less than 5!

Numbers less than 5 are to the left of 5 on the number line. 051015-20-15-10-5-252025 If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right. There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc. The number 5 would not be a correct answer, though, because 5 is not less than 5. The open dot means that 5 is not a solution The number line is shaded to the left of 5, so all numbers less than 5 are solutions

“x ≥ -2” means that whatever value x has, it must be greater than or equal to -2. Try to name ten numbers that are greater than or equal to - 2!

Numbers greater than -2 are to the right of 5 on the number line. 051015-20-15-10-5-252025 If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right. There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. The number -2 would also be a correct answer, because of the phrase, “or equal to”. -2 The closed dot means that -2 is a solution The number line is shaded to the right of -2, so all numbers greater than -2 are solutions

Where is -1.5 on the number line? Is it greater or less than -2? 051015-20-15-10-5-252025 -1.5 is between -1 and -2. -1 is to the right of -2. So -1.5 is also to the right of -2. -2

Solve an Inequality w + 5 < 8 We will use the same steps that we did with equations, if a number is added to the variable, we add the opposite sign to both sides: w + 5 + (-5) < 8 + (-5) w + 0 < 3 w < 3 All numbers less than 3 are solutions to this problem!

More Examples 8 + r ≥ -2 8 + r + (-8) ≥ -2 + (-8) r + 0 ≥ -10 w ≥ -10 All numbers from -10 and up (including -10) make this problem true!

More Examples x - 2 > -2 x + (-2) + (2) > -2 + (2) x + 0 > 0 x > 0 All numbers greater than 0 make this problem true!

More Examples 4 + y ≤ 1 4 + y + (-4) ≤ 1 + (-4) y + 0 ≤ -3 y ≤ -3 All numbers from -3 down (including -3) make this problem true!

3.3 Solving Inequalities by Multiplying or Dividing Objective: Solve inequalities by using the multiplication and division properties of inequalities.

5-Minute Check Solve each inequality. 1.r – 9 < 4 2.s + 24 > 23 3.t – (-30)  40 4.u + (-14)  -29 5.19 28

3.3 Solving Inequalities by Multiplying or Dividing When you multiply or divide each side of an inequality by a positive integer, the result remains true.

3.3 Solving Inequalities by Multiplying or Dividing Example 1 Solve 9x > -36 and graph the solution on a number line.

3-7 Solving Inequalities by Multiplying or Dividing Example 1 Solve 9x > -36 and graph the solution on a number line. 9x > -36

3-3 Solving Inequalities by Multiplying or Dividing Example 1 Solve 9x > -36 and graph the solution on a number line. 9x > -36 9x/9 > -36/9Divide each side by 9.

3-3 Solving Inequalities by Multiplying or Dividing Example 1 Solve 9x > -36 and graph the solution on a number line. 9x > -36 9x/9 > -36/9Divide each side by 9. x > -4

3-3 Solving Inequalities by Multiplying or Dividing Example 1 Solve 9x > -36 and graph the solution on a number line. 9x > -36 9x/9 > -36/9Divide each side by 9. x > -4

Notice that -2 is included

3-3 Solving Inequalities by Multiplying or Dividing Example 2 Solve -4x  12 and graph the solution on a number line.

3-3 Solving Inequalities by Multiplying or Dividing Example 2 Solve -4x  12 and graph the solution on a number line. -4x  12

3-3 Solving Inequalities by Multiplying or Dividing Example 2 Solve -4x  12 and graph the solution on a number line. -4x  12 -4x/(-4)  12/(-4)Divide each side by –4 and reverse the order symbol.

3-3 Solving Inequalities by Multiplying or Dividing Example 2 Solve -4x  12 and graph the solution on a number line. -4x  12 -4x/(-4)  12/(-4)Divide each side by –4 and x  -3 reverse the order symbol.

3-3 Solving Inequalities by Multiplying or Dividing Example 2 Solve -4x  12 and graph the solution on a number line. -4x  12 -4x/(-4)  12/(-4)Divide each side by –4 and x  -3 reverse the order symbol.

Question? How is the inequality symbol related to the shading on the number line? When the inequality is written with variable on the left, the inequality symbol points in the direction of the shading.

3-3 Solving Inequalities by Multiplying or Dividing Assignment: do c, e, f in your class folder You must solve, check your answer

3.4 solve and Graph Multi-Step Inequalities

Classwork Complete Solving Inequalities pages 1 -2 in your classwork folder. You must solve, check and graph