Algebra 1 > Notes > Yorkcounty Final > Unit 4 > Lesson_9___solving_inequalities

Solving Inequalities

Lesson 9

What does Inequality mean? 

 



An inequality is something that looks like < , > , < or >. x = 5 means that x equals 5 only. x < 5 means that x is less than 5. This means that x is NOT equal to 5 but it is any number less. 4.9999, 3, 0, -43, -83.213, etc. x < 5 means that x is less than OR equal to 5. So x CAN equal 5 and any number less. 5, 4.99, 3, 0 , -43, etc.

Inequality Signs x < 9, x is less than 9 x > 9, x is greater than 9 x < 9, x is less than or equal to 9 x > 9, x is greater than or equal to 9

Multiple Meanings 9

< x , this means that 9 is less than x. Which could also be read as x is greater than 9.  1st meaning – 9 is less than x. If 9 is less than x then x must be larger than 9.  2nd meaning – x is greater than 9.  9 < x and x > 9 mean the exact same thing!

Showing Inequality 

 

Graphing on a number line is a way of expressing the answer to an inequality. x < 3, means that x is any number less than 3 but NOT 3. First you plot a point on 3. You use an open circle here at 3 on the number line. You use an open circle when you have inequality signs like < or >. After that you shade in the appropriate direction. Since our solution is any number less than 3, we will shade to the left.

-4

-3

-2

-1

0

1

2

3

4

Showing Inequality • How do we show x < 3? This reads x is less than or equal to 3. •Remember last time we had x < 3, this reads x is less than 3. We plotted an open circle at 3. This time we have x < 3, x is less than or equal to 3. Now we are going to use a closed circle at 3 to represent this. After that point is plotted shade in the appropriate direction. Since it says x is less than 3 we will shade where the numbers are less than 3. To the left.

-4

-3

-2

-1

0

1

2

3

4

Compound Inequality Perhaps x can be greater than -2 and less than 4. How can we express that? x > -2, x is greater than 2. (NOTE: x > -2 and -2 < x mean the same thing!) x < 4, x is less than 4. -2 < x < 4, x is greater than -2 and x is less than 4.

-4

-3

-2

-1

0

1

2

3

4

Compound Inequalities What does 0 < x < 3 mean? 0 < x means that x is greater than or equal to 0 and x < 3 means that x is less than 3. x is greater than or equal to 0. Closed circle on 0, pointing to the right. x is less than 3. Open circle on 3, pointing to the left.

-4

-3

-2

-1

0

1

2

3

4

Solving Linear Inequalities  Solve

3x < 9, get x by itself on the left side.  Divide both sides by 3 and you get.  x < 3. What does this answer mean?  x is less than 3 means x is any number less than 3 but NOT 3. So x can be 2.9, 2.9 < 3, that’s true. What about x = -3. -3 < 3. That is true also.

Solving Inequalities  Solve  Try

for x: 3x + 1 > 4

this one on your own.  Go to the next slide to see the solution.

Solving Inequalities   

 

3x + 1 > 4 3x > 3 x>1

First, subtract one from both sides. Next divide both sides by 3. x is greater than or equal to one is the solution.

x = 1, 1 > 1, that is true ( 1 is greater than or equal to 1) x = 9, 9 > 1, that is true also (9 is greater than or equal to 1)

Dividing or Multiplying by a Negative There is a special rule that happens when you divide or multiply by a negative number. When you do this you must FLIP the inequality. Example: Solve for x -2x < 4 (divide both sides by -2) x>2 (notice I flipped the inequality) Check your answer. x is greater than 2. Let x be equal to 4. Plug into the original equation. -2(4) < 4 -8 < 4, that’s true

Dividing or Multiplying by a negative    

 

Solve for x: -x + 5 < 7 -x + 5 < 7 (subtract 5 from both sides) -x < 2 (-1 is being multiplied to x, so divide -1 from both sides) x > -2 (NOTE: divide by a negative, flip the inequality) Check the solution: x is greater than -2. Let x be -1. Substitute -1 for x. -(-1) + 5 < 7 1+5< 7 6 < 7, that’s true

Solving Compound Inequalities  Solve

for x: 3x + 1 < 10 < x + 8  First write it out as two problems  1) 3x + 1 < 10  2) 10 < x + 8  Solve both problems

Solving Compound Inequalities 3x + 1 < 10 3x < 9 x<3

(subtract 1 from both sides) ( divide both sides by 3) (x is less than 3)

10 < x +6 4<x

(subtract 6 from both sides) (x is greater than 4)

-4

-3

-2

-1

0

1

2

3

4

Practical Applications A

soda costs at most $2.00  We can represent that as s < 2 (s = soda)  The

soda can cost less than 2 and at most 2.

Practical Applications A

car costs 100 dollars to rent and 2 dollars for every mile you drive. How many miles can you drive if you have 183 dollars?  100 dollars plus two dollars a mile is less than or equal to 183. (m = miles)  100 + 2m < 183 Now solve for m

Practical Applications



100 + 2m < 183 2m < 83 m < 41.5



This means you can drive 41.5 miles and be under your budget.

 

(subtract 100 from both sides) (divide by 2)

Practical Applications 

Company X make 18 dollars profit from every computer it sells. They need to make at least 450 dollars profit a week to make their quota. How many computer must they sell to make their quota?



Write an inequality for computers sold. They make 18 dollars for each computer sold and they need to make at least 450 dollars profit a week.



Practical Applications    

 

18c > 450 Solve for c 18c > 450 c > 25

(18 times c means 18 dollars per computer sold) (divide both sides by 8)

What does this mean? Company X needs to sell at least 25 computers a week to make their quota.

Example  Solve

for x: -3x + 10 < 22  Graph solution on a number line.

Example -3x + 10 < 22 -3x < 12

(subtract 10 from both sides) (divide both sides by negative 3)



x > -4

(divide by a negative, flip inequality)



Graph solution on a number line.

 

Example x > -4

Open circle on -4, shading to the right

-4

-3

-2

-1

0

1

2

3

4