Solving and Graphing Inequalities CHAPTER 6 REVIEW Solving and Graphing Inequalities

≥ : 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

“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. 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.

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!

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”.

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.

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

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!

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!

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!

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.

Example: -3y > 18 Solve: -3y + 5 >23 -5 -5 -5 -5 -3y > 18 -3 -3 y < -6 Subtract 5 from each side. Divide each side by negative 3. Reverse the inequality sign. Graph the solution. -6

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 -3 -3 x > 2 + 11 + 11 -c > 34 -1 -1 c < -34 3r – 6 < 2r + 4 -2r -2r r – 6 < 4 +6 +6 r < 10

You did remember to reverse the signs . . . Good job! . . . didn’t you?

Ring the alarm! We divided by a negative! Example: - 4x - 4x Ring the alarm! We divided by a negative! + 6 +6 -2 -2 We turned the sign!

Remember Absolute Value

* Plug in answers to check your solutions! Ex: Solve 6x-3 = 15 6x-3 = 15 or 6x-3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

Get the abs. value part by itself first! Ex: Solve 2x + 7 -3 = 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.

Ex: Solve & graph. Becomes an “and” problem -3 7 8

Solve & graph. Get absolute value by itself first. Becomes an “or” problem -2 3 4

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

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

Absolute Value Inequalities Case 1 Example: and

Absolute Value Inequalities Case 2 Example: or OR

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

Graphing Linear Inequalities in Two Variables SWBAT graph a linear inequality in two variables SWBAT Model a real life situation with a linear inequality.

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

When dealing with slanted lines If it is > or  then you shade above If it is < or  then you shade below the line

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.

Graphing a Linear Inequality Sketch a graph of y  3

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