## Solving Inequalities

Sometimes we need to solve Inequalities like these:

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

We call that "solved".

## Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

x > 10

## How to Solve

Solving inequalities is very like solving equations , we do most of the same things ...

... but we must also pay attention to the direction of the inequality .

Some things can change the direction !

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

## Safe Things To Do

These things do not affect the direction of the inequality:

- Add (or subtract) a number from both sides
- Multiply (or divide) both sides by a positive number
- Simplify a side

## Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

But these things do change the direction of the inequality ("<" becomes ">" for example):

- Multiply (or divide) both sides by a negative number
- Swapping left and right hand sides

## Example: 2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality :

12 > 2y+7

Here are the details:

## Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra ), like this:

## Example: x + 3 < 7

If we subtract 3 from both sides, we get:

x + 3 − 3 < 7 − 3

And that is our solution: x < 4

In other words, x can be any value less than 4.

## What did we do?

And that works well for adding and subtracting , because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

## What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

## Example: 12 < x + 5

If we subtract 5 from both sides, we get:

12 − 5 < x + 5 − 5

That is a solution!

But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

Do you see how the inequality sign still "points at" the smaller value (7) ?

And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

## Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying ).

But we need to be a bit more careful (as you will see).

## Positive Values

Everything is fine if we want to multiply or divide by a positive number :

## Example: 3y < 15

If we divide both sides by 3 we get:

3y /3 < 15 /3

And that is our solution: y < 5

## Negative Values

Well, just look at the number line!

For example, from 3 to 7 is an increase , but from −3 to −7 is a decrease.

See how the inequality sign reverses (from < to >) ?

Let us try an example:

## Example: −2y < −8

Let us divide both sides by −2 ... and reverse the inequality !

−2y < −8

−2y /−2 > −8 /−2

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

## Multiplying or Dividing by Variables

Here is another (tricky!) example:

## Example: bx < 3b

It seems easy just to divide both sides by b , which gives us:

... but wait ... if b is negative we need to reverse the inequality like this:

But we don't know if b is positive or negative, so we can't answer this one !

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b :

- if b is 1 , then the answer is x < 3
- but if b is −1 , then we are solving −x < −3 , and the answer is x > 3

The answer could be x < 3 or x > 3 and we can't choose because we don't know b .

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

## A Bigger Example

Example: x−3 2 < −5.

First, let us clear out the "/2" by multiplying both sides by 2.

Because we are multiplying by a positive number, the inequalities will not change.

x−3 2 ×2 < −5 ×2

x−3 < −10

Now add 3 to both sides:

x−3 + 3 < −10 + 3

And that is our solution: x < −7

## Two Inequalities At Once!

How do we solve something with two inequalities at once?

## Example: −2 < 6−2x 3 < 4

First, let us clear out the "/3" by multiplying each part by 3.

Because we are multiplying by a positive number, the inequalities don't change:

−6 < 6−2x < 12

−12 < −2x < 6

Now divide each part by 2 (a positive number, so again the inequalities don't change):

−6 < −x < 3

Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction .

6 > x > −3

And that is the solution!

But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

−3 < x < 6

- Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
- Multiplying or dividing both sides by a negative number
- Don't multiply or divide by a variable (unless you know it is always positive or always negative)

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## Ninth Grade (Grade 9) Inequalities Questions

You can create printable tests and worksheets from these Grade 9 Inequalities questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.

- [math]x > -5[/math]
- [math]x>=-5[/math]
- [math]x<-5[/math]
- [math]x<=-5[/math]
- [math]x >= -2/3[/math]
- [math]x > 2/3[/math]
- [math]x > -3/2[/math]
- [math]x >= 3/2[/math]
- [math]x>4[/math]
- [math]x<=6[/math]
- [math]x>=4[/math]
- [math]x<4[/math]
- [math]x> -4[/math]
- [math]x > 4[/math]
- [math]x < -4[/math]
- [math]x < 4[/math]
- [math]x < -8[/math]
- [math]x > -8[/math]
- [math]x > 8[/math]
- [math]x < 8[/math]
- [math]x>=-2 1/3[/math]
- [math]x>=3[/math]
- [math]x<=3[/math]
- [math]x>=2 1/3[/math]
- none of the above
- [math]x<5[/math]
- [math]x<=5[/math]
- [math]x>5[/math]
- [math]x>=5[/math]
- [math]x<=-4[/math]
- [math]x<=4[/math]
- [math]x < -3/4[/math]
- [math]x > -3/4[/math]
- [math]x < 3/4[/math]
- [math]x > 3/4[/math]
- [math]x < -3 [/math]
- [math]x < 3 [/math]
- [math]x > -3[/math]
- [math]x > 3[/math]
- x > -2/7
- x < -2/7
- x > -10/7
- x < -10/7
- [math]p<=8[/math]
- [math]p>=8[/math]
- [math]p>8[/math]
- [math]w<7[/math]
- [math]w<2[/math]
- [math]w<-7[/math]
- [math]w<-2[/math]
- [math]w>20[/math]
- [math]w<-5[/math]
- [math]w>5[/math]
- [math]x<-4[/math]
- [math]7< f <11[/math]
- [math]f<7 or f>11[/math]
- [math]7<=f<=11[/math]
- [math]f<=7 or f>=11[/math]
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## Inequalities

In this inequalities worksheet, 9th graders solve and complete 25 various types of problems. First, they determine the portion of the graph that should be shaded for each inequality. Then, students graph each system of inequalities. In addition, they determine which graph best represents each inequality.

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Graphing systems, do these systems meet your expectations, systems of equations and inequalities, systems of linear inequalities, graphical system of inequalities, solving systems of equations three ways, graphing systems of equations, richland com. college: graphing systems of inequalities, systems of equations, desmos graphing calculator.

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3 ≤ x ≤ 10. 10x + 4 ≥ -96 and x + 9 ≤ 3. -10 ≤ x ≤ -6. -7 - 8x ≥ 41 or 3x - 3 ≥ -12. x ≤ -6 or x ≥ -3. -8 - 3x < -29 or 6x - 4 < 32. x > 7 or x < 6. Study with Quizlet and memorize flashcards containing terms like -91 ≥ 7x, 30 > 15x, -8 - x ≥ 5 and more.

-1 1 1.5 0 1.5 Identify the property used in each step of solving the inequality . Step 2: Addition property Step 3: Multiplication property Which numbers are solutions to the inequality 2x - 1 > x + 2? Check all of the boxes that apply.

1 / 5 Flashcards Learn Test Match Q-Chat Created by loringreen Terms in this set (5) What inequality represents the sentence? The product of a number and 12 is no more than 15. 12n≤15 We have an expert-written solution to this problem! 26+6b≥2 (3b+4) all real numbers A club decides to sell T-shirts for $15 as a fundraiser.

1 / 19 Flashcards Learn Test Match Q-Chat Created by Jacquelyn_Fischvogt Teacher How to graph and solve inequalities on a number line. Terms in this set (19) n<-1 Solve: 2n+6<4 n>3 Solve: 5n-4>11 x<1/2 Solve: 2x+5<6 x>-3 Solve: 7x>-21 x>4 Solve: x + 6 > 10 open When graphing < and > uses what type of circle: open or closed? closed

How to Solve Solving inequalities is very like solving equations, we do most of the same things ... ... but we must also pay attention to the direction of the inequality. Direction: Which way the arrow "points" Some things can change the direction! < becomes > > becomes < ≤ becomes ≥ ≥ becomes ≤ Safe Things To Do

Assignment # 9: Writing and Solving Systems of Inequalities Name: 1 Jordan works for a landscape company during his summer vacation. He is paid $12 per hour for mowing lawns and $14 per hour for planting gardens. He can work a maximum of 40 hours per week, and would like to earn at least $250 this week.

For problems 1 - 10 solve each of the following inequalities. Give the solution in both inequality and interval notations. 7x+2(4−x) < 12−3(5+6x) 7 x + 2 ( 4 − x) < 12 − 3 ( 5 + 6 x) 10(3+w) ≥ 9(2 −4w) 10 ( 3 + w) ≥ 9 ( 2 − 4 w) 2(4 +5y) ≤ 12y −6(1−3y) 2 ( 4 + 5 y) ≤ 12 y − 6 ( 1 − 3 y)

Unit 1 Algebra foundations Unit 2 Solving equations & inequalities Unit 3 Working with units Unit 4 Linear equations & graphs Unit 5 Forms of linear equations Unit 6 Systems of equations Unit 7 Inequalities (systems & graphs) Unit 8 Functions Unit 9 Sequences Unit 10 Absolute value & piecewise functions Unit 11 Exponents & radicals

Algebra (all content) 20 units · 412 skills. Unit 1 Introduction to algebra. Unit 2 Solving basic equations & inequalities (one variable, linear) Unit 3 Linear equations, functions, & graphs. Unit 4 Sequences. Unit 5 System of equations. Unit 6 Two-variable inequalities. Unit 7 Functions. Unit 8 Absolute value equations, functions, & inequalities.

solution any value of a variable that turns an open sentence into a true statement −5−3 (g+7)<g+3 g>−7.25 Four people are going as a group to watch a movie. A movie ticket costs $9. A snack box costs $3. They have a total of $50. The inequality 36+3b≤50 can be used to determine how many snack boxes, b, they can buy.

Study with Quizlet and memorize flashcards containing terms like Which interval is the solution set to 0.35x - 4.8 < 5.2 - 0.9x?, Complete the steps to solve the inequality: 0.2(x + 20) - 3 > -7 - 6.2x Use the distributive property: Combine like terms: Use the addition property of inequality: Use the subtraction property of inequality: Use the division property of inequality: 0.2x + 4 - 3 > -7 ...

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Write an inequality for each sentence. An assignment requires less than 35 minutes. x>35. Write an inequality for each sentence. An assignment requires more than 35 minutes. ... Solve. 9≤x/14+6. Other sets by this creator. Value of Money. 29 terms. chelsie_l_sturdivant. Vocabulary Practice. 92 terms. chelsie_l_sturdivant.

Curated and Reviewed by Lesson Planet. In this inequalities worksheet, 9th graders solve and complete 32 different problems that include using multiplication and division in each. First, they solve each given inequality. Then, students check their solutions, paying close attention to the signs. They also define a variable, write an inequality ...

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In this section we will be solving (single) inequalities that involve polynomials of degree at least two. Or, to put it in other words, the polynomials won't be linear any more. Just as we saw when solving equations the process that we have for solving linear inequalities just won't work here.

Chuck Towle. The equation y>5 is a linear inequality equation. y=0x + 5. So whatever we put in for x, we get x*0 which always = 0. So for whatever x we use, y always equals 5. The same thing is true for y>5. y > 0x + 5. And again, no matter what x we use, y is always greater than 5.

Here are a set of assignment problems for the Solving Equations and Inequalities chapter of the Algebra notes. Please note that these problems do not have any solutions available. These are intended mostly for instructors who might want a set of problems to assign for turning in.

Answer. Exercise 9.7.4. Solve and write the solution in interval notation: 3x x − 4 < 2. Answer. In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator. Example 9.7.3. Solve and write the solution in interval notation: 5 x2 − 2x − 15 > 0.

08. hr. min. sec. SmartScore. out of 100. IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)!

Then, students solve each system of inequalities and the area where the shadings from each inequality overlap one another is the answer. 5 Views 19 Downloads Concepts. negative integers, positive integers, negative numbers, positive numbers, basic operations, like terms, simplifying expressions, systems of inequalities, inequalities.

Solve the following inequality. 2x < 8. x<8. x<4. x>4. x>8. Grade 9 Inequalities. Which of the following is NOT A SOLUTION to the following inequality? − 2x + 3 ≤ 9 - 2 x + 3 ≤ 9.

In this inequalities worksheet, 9th graders solve and complete 25 various types of problems. First, they determine the portion of the graph that should be shaded for each inequality. Then, students graph each system of inequalities. In addition, they determine which graph best represents each inequality.