Solving Inequalities Word Problems

Inequalities, Word Problems, Slope

By Shalynne Orth

Inequalities

What is an inequality?

How are equations and inequalities different?

Equations

Have an = sign

Only have one answer when solved

Inequalities

Don't have and = sign instead use inequality symbols

< (less than)

> (greater than)

≤ (less than or equal to)

≥ (greater than or equal to)

Can have multiple answers when solved

Example of an inequality and how to interpret your answer.

Multiple Choice

Solve the following inequality:

x + 2 > 12

• subtract 2 from both sides and your answer is

  x = 10

• subtract 2 from both sides and your answer is

  x > 10

• subtract 2 from both sides and your answer is

  X < 10

Multiple Choice

Solve 5x > 10

• divide by 5 on both sides and your answer is

  x = 2

• divide by 5 on both sides and your answer is

  x < 2

• divide by 5 on both sides and your answer is

  x > 2

Multiple Choice

Solve 3x + 2 > 8

• subtract 2 from both sides and your answer is

  x > 6

• subtract 2 from both sides and your answer is

  x = 6

• subtract 2 from both sides and then divide by 3 on both sides and your answer is

  x > 2

One Rule: If you are multiplying or dividing by a negative number you must switch the direction of the inequality symbol. 

• (ex) Solve -4x > -12

• Divide both sides by -4, and because we are dividing by a negative number, we must reverse the inequality sign

  ⇒ -4x/-4 < -12/-4

  ⇒ x < 3

Multiple Choice

Try some:

-2x ≤ 4

• divide by -2 o both sided and your answer will be

  x = -2

• divide by -2 on both sides and you will switch the direction of the inequality since you are dividing by a negative and your answer will be

  x ≥ -2

• divide by -2 on both sides and your answer will be

  x ≤ -2

Multiple Choice

- 3x < -12

• divide both sides by -3 and your answer is

  x < 4

• x = 4

• divide both sides by -3 and your answer is

  x > 4

  because you divided by a negative and had to flip the inequality symbol

Multiple Choice

Solve  -9h > -63

• h < 7
• h > 7
• h < -7
• h > -7

Things to Remember:

-The inequalities that contains the equal signs are closed circles when modeling on the number line.

-The inequalities without the equal sign are open circles when modeling on the number line.

Multiple Choice

Decide if the inequality is open circle or closed circle: 4x > 16

• Open Circle

• Closed Circle

• Flip the sign

• I don't know

Multiple Choice

Which inequality is represented by the following graph

• x < -8

• x > -8

• x ≥ -8

• x ≤ -8

Multiple Choice

Match the graph with its inequality.

• a < 11

• a > 11

• a ≥ 11

• a ≤ 11

Multiple Choice

• x is less than or equal to -2

• x < -2

• -2 < x

• x is greater than -2

Multiple Choice

Solve and choose the correct graph:

• A
• B
• C
• D

Writing and Graphing Compound Inequalities

Objective: Write and graph compound inequalities.

Multiple Choice

Which compound inequality represents:  $$x<4$$  and  $$x\ge-10$$  

• $$-10<x\le4$$  

• $$-10<x<4$$

• $$-10\le x\le4$$

• $$-10\le x<4$$

Multiple Choice

Which graph represents the compound inequality  $$-5<x<15$$  ?

Multiple Select

Which inequalities represent the same thing?

• $$x\le7$$ and $$x>3$$

• $$3\le x<7$$

• $$x<3$$ and $$x>7$$

• $$3<x<7$$

• $$3<x\le7$$

​Graphing "or" inequalities:

​

Just graph two inequalities on the same number line!​

Multiple Choice

Write an inequality for Graph 3.

• x ≤ -2 AND x > 5
• x ≤ -2 OR x > 5
• x ≥ -2 OR x < 5
• -2 ≤ x < 5

Multiple Choice

Which graph represents x<-2 or x>3?

• Graph 1

• Graph 2

• Graph 3

Multiple Select

Which inequalities represent the graph?

_______ or _______

• $$x\ge-10$$

• $$x\le-10$$

• $$x>20$$

• $$x<20$$

• $$-10\le x<20$$

word problems

writing equations

Multiple Choice

WRITE THE EQUATION FOR THIS PROBLEM:

There are 45 students on the football team. This is 13 more than the number of students on the basketball team. How many students are on the basketball team?

• 45 = 13 + x

• 45 + 13 =x

• 43 + 13 = 56

Multiple Choice

A used mountain bike costs $79.95. This is $120 less than the cost of a new one. If c is the cost of the new bike, which equation can you use to find the cost of a new bike?

• 120 - 79.95 = c

• 79.95 = c -120

Multiple Choice

The goal of the School Bake Sale is to raise $125 more than last year’s sale. Last year the Bake Sale raised $320. What is the goal this year?

• x = 125 + 320

• x + 125 = 320

Inequalities Word Problems

Writing Inequalities from Word Problems

When solving inequality word problems, remember the key words and phrases that go along with each inequality symbol:

Multiple Choice

The hole for a birdfeeder post is 3 feet deep. The top of the post needs to be at least 5 feet above the ground.

Which of the following inequalities would correctly represent the required length of the post?

• $$x\ \ge\ 5$$

• $$x\ \le\ 5$$

• $$x\ -\ 3\ \ge\ 5$$

• $$x\ +\ 3\le\ 5$$

Multiple Choice

You have $30 to buy baseball cards. Each pack of cards costs $5. Write an inequality that represents the number of packs of baseball cards you can buy and still have at least $10 left.

• $$30\ -\ 5x\ \ge10$$

• $$30\ -\ 5x\ \le\ 10$$

• $$30\ -\ 5x\ <\ 10$$

• $$30\ -\ 5x\ >\ 10$$

Slope

​

• Identifying slope (direction)

• Finding slope of a line (steepness)

• Finding slope from two points​

• Slope from a Table

Identifying Slope of a Line

• Slope can tell us the direction of the line!

• Refresh yourself of the 4 types of slope using the picture to the right:

Finding the Slope of a Line

• Slope can tell us how steep a line is!

• Each line can be assigned a value (a #) that tells us how steep the line is.

• Slope is written as a ratio (rise/run)

• Always make sure to simplify your answer!

Multiple Choice

Find the slope of the line below using rise/run.

• $$\frac{2}{3}$$

• $$\frac{3}{2}$$

• 3

• $$\frac{1}{3}$$

Multiple Choice

Find the slope of the graph. ( Slopes must be written as a fraction and simplified if possible)

• 3/4

• -3/4

• -4/3

• 4/3

You can use any two points on a line to find the slope

How to Find Slope From Two Points

Follow the steps shown:

Fill in the Blank

Find the slope: (-1, 8) (0, 6)

Fill in the Blank

Find the slope: (- 3, - 5) (- 7, - 4)

Finding slope from a Table

• Slope = change in y/change in x

• Use the table to find the value the y is changing by. That is the numerator!

• Use the table to find the value the x is changing by. That is the denominator!

• ANOTHER OPTION: Graph the points and count rise/run

Multiple Choice

Find the slope. Simplify, if you can.

• $$\frac{2}{5}$$

• $$\frac{5}{2}$$

• $$2\ \frac{1}{2}$$

• 2

Multiple Choice

Find the slope. Simplify, if you can.

• 3

•   $$\frac{1}{3}$$  

• $$-\frac{1}{3}$$  

• -3