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**Solving and Graphing Inequalities**

CHAPTER 6 REVIEW Solving and Graphing Inequalities

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**≥ : greater than or equal to**

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

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**“x < 5” means that whatever value x has, it must be less than 5.**

Try to name ten numbers that are less than 5!

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**Numbers less than 5 are to the left of 5 on the number line.**

5 10 15 -20 -15 -10 -5 -25 20 25 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.

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**Try to name ten numbers that are greater than or equal to -2!**

“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!

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**Numbers greater than -2 are to the right of 5 on the number line.**

5 10 15 -20 -15 -10 -5 -25 20 25 -2 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”.

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**Where is -1.5 on the number line? Is it greater or less than -2?**

5 10 15 -20 -15 -10 -5 -25 20 25 -2 -1.5 is between -1 and -2. -1 is to the right of -2. So -1.5 is also to the right of -2.

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**All numbers less than 3 are solutions to this problem!**

Solve an Inequality w + 5 < 8 w (-5) < 8 + (-5) All numbers less than 3 are solutions to this problem! w < 3

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**All numbers from -10 and up (including -10) make this problem true!**

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

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**All numbers greater than 0 make this problem true!**

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

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**All numbers from -3 down (including -3) make this problem true!**

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

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**There is one special case.**

Sometimes you may have to reverse the direction of the inequality sign!! That only happens when you multiply or divide both sides of the inequality by a negative number.

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**Example: -3y > 18 Solve: -3y + 5 >23 -5 -5**

-3y > 18 y < -6 Subtract 5 from each side. Divide each side by negative 3. Reverse the inequality sign. Graph the solution. -6

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**Try these: 1.) Solve 2x + 3 > x + 5 2.)Solve - c – 11 >23**

3.) Solve 3(r - 2) < 2r + 4 -x x x + 3 > 5 x > 2 -c > 34 c < -34 3r – 6 < 2r + 4 -2r r r – 6 < 4 r < 10

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**You did remember to reverse the signs . . .**

Good job! . . . didn’t you?

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**Ring the alarm! We divided by a negative!**

Example: - 4x x Ring the alarm! We divided by a negative! We turned the sign!

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**Remember Absolute Value**

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*** Plug in answers to check your solutions!**

Ex: Solve 6x-3 = 15 6x-3 = or 6x-3 = -15 6x = or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

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**Get the abs. value part by itself first!**

Ex: Solve 2x = 8 Get the abs. value part by itself first! 2x+7 = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11 2x = 4 or 2x = -18 x = 2 or x = -9 Check the solutions.

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Ex: Solve & graph. Becomes an “and” problem

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**Solve & graph. Get absolute value by itself first.**

Becomes an “or” problem

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**Example 1: |2x + 1| > 7 2x + 1 > 7 or 2x + 1 >7**

This is an ‘or’ statement. (Greator). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. |2x + 1| > 7 2x + 1 > 7 or 2x + 1 >7 2x + 1 >7 or 2x + 1 <-7 x > 3 or x < -4 3 -4

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**Example 2: This is an ‘and’ statement. (Less thand). Rewrite.**

In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. |x -5|< 3 x -5< 3 and x -5< 3 x -5< 3 and x -5> -3 x < 8 and x > 2 2 < x < 8 8 2

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**Absolute Value Inequalities**

Case 1 Example: and

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**Absolute Value Inequalities**

Case 2 Example: or OR

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**Solutions: No solution**

Absolute Value Answer is always positive Therefore the following examples cannot happen. . . Solutions: No solution

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**Graphing Linear Inequalities in Two Variables**

SWBAT graph a linear inequality in two variables SWBAT Model a real life situation with a linear inequality.

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**Some Helpful Hints If the sign is > or < the line is dashed**

If the sign is or the line will be solid When dealing with just x and y. If the sign > or the shading either goes up or to the right If the sign is < or the shading either goes down or to the left

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**When dealing with slanted lines **

If it is > or then you shade above If it is < or then you shade below the line

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**Graphing an Inequality in Two Variables**

Graph x < 2 Step 1: Start by graphing the line x = 2 Now what points would give you less than 2? Since it has to be x < 2 we shade everything to the left of the line.

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**Graphing a Linear Inequality**

Sketch a graph of y 3

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**Step 1: Put into slope intercept form**

Using What We Know Sketch a graph of x + y < 3 Step 1: Put into slope intercept form y <-x + 3 Step 2: Graph the line y = -x + 3

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