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## Unit 2: Lesson 5

- Inequalities with variables on both sides
- Inequalities with variables on both sides (with parentheses)
- Multi-step inequalities
- Using inequalities to solve problems

## Multi-step linear inequalities

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## Section 2.11 : Linear Inequalities

- \(4\left( {z + 2} \right) - 1 > 5 - 7\left( {4 - z} \right)\) Solution
- \(\displaystyle \frac{1}{2}\left( {3 + 4t} \right) \le 6\left( {\frac{1}{3} - \frac{1}{2}t} \right) - \frac{1}{4}\left( {2 + 10t} \right)\) Solution
- \( - 1 Solution
- \(8 \le 3 - 5z Solution
- \(0 \le 10w - 15 \le 23\) Solution
- \(\displaystyle 2 Solution
- If \(0 \le x Solution

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## Solving Inequalities Practice Problems

Let's quickly recap some of the steps for solving these practice problems.

## Steps for Solving Inequalities

- Solve the inequality as you would an equation which means that "whatever you do to one side, you must do to the other side".
- If you multiply or divide by a negative number, REVERSE the inequality symbol.

## Steps for Graphing Your Solution to the Inequality

- Use an open circle on the graph if your inequality symbol is greater than or less than.
- Use a closed circle on the graph if your inequality symbol is greater than or equal to OR less than or equal to.
- Arrow will point to the left if the inequality symbol is less than .
- Arrow will point to the right if the inequality symbol is greater than .

## Take a look at the questions that other students have submitted:

Another compound inequality problem

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## 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:

## Example: x + 2 > 12

## 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 !

## 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):

## Example: 2y+7 < 12

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

## Adding or Subtracting a Value

## Example: x + 3 < 7

If we subtract 3 from both sides, we get:

And that is our solution: x < 4

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

## What did we do?

## 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:

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:

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 >) ?

## Example: −2y < −8

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

And that is the correct solution: y > 4

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

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 .

## A Bigger Example

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.

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:

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

- 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)

## Solving Inequalities

Related Pages Solving Equations Algebraic Expressions More Algebra Lessons

In these lessons, we will look at the rules, approaches, and techniques for solving inequalities.

To solve an inequality, we can:

- Add the same number to both sides.
- Subtract the same number from both sides.
- Multiply both sides by the same positive number.
- Divide both sides by the same positive number.
- Multiply both sides by the same negative number and reverse the sign.
- Divide both sides by the same negative number and reverse the sign.

## Inequalities Of The Form “x + a > b” or “x + a < b”

Solution: x + 7 < 15 x + 7 – 7 < 15 – 7 x < 8

## Inequalities Of The Form “x – a < b” or “x – a > b”

Solution: x – 6 > 14 x – 6+ 6 > 14 + 6 x > 20

Example: Solve the inequality x – 3 + 2 < 10

Solution: x – 3 + 2 < 10 x – 1 < 10 x – 1 + 1 < 10 + 1 x < 11

## Inequalities Of The Form “a – x < b” or “a – x > b”

Example: Solve the inequality 7 – x < 9

Example: Solve the inequality 12 > 18 – y

## Inequalities Of The Form “ < b” or “ > b”

Solving linear inequalities with like terms.

Example: Evaluate 3x – 8 + 2x < 12

Solution: 3x – 8 + 2x < 12 3x + 2x < 12 + 8 5x < 20 x < 4

Example: Evaluate 6x – 8 > x + 7

Solution: 6x – 8 > x + 7 6x – x > 7 + 8 5x > 15 x > 3

Example: Evaluate 2(8 – p) ≤ 3(p + 7)

## An Introduction To Solving Inequalities

## Solving One-Step Linear Inequalities In One Variable

## Solving Two-Step Linear Inequalities In One Variable

## Solving Linear Inequalities

Examples of how to solve linear inequalities are shown:

Example: Solve: 3x - 6 > 8x - 7

For example, to solve -3x is less than 12, divide both sides by -3, to get x is greater than -4.

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## SAT Math : Inequalities

What is the range of values for y, expressed in terms of x?

Recall that with absolute values and "less than" inequalities, we have to hold the following:

In this form, we can solve for y. First, we have to subtract x from all 3 parts of the inequality:

Now, we have to divide each element by 3:

(–15 – 12x)/3 < y < (15 – 12x)/3

What is a possible valid value of x?

This inequality could be rewritten as:

4x + 14 < –30; 4x < –44; x < –11

Therefore, anything between –11 and 4 (inclusive) will not work. Hence, the answer is 7.

## Example Question #3 : Inequalities

Given the inequality, |2 x – 2| > 20,

what is a possible value for x ?

For this problem, we must take into account the absolute value.

A possible value for x would be –10 since that is less than –9.

## Example Question #4 : Inequalities

## Example Question #5 : Inequalities

## Example Question #131 : Algebra

Subtract 5 from both sides of the inequality:

Therefore only I must be true.

## Example Question #7 : Inequalities

Solve for both x – 3 < 2 and –( x – 3) < 2.

The results are x < 5 and x > 1.

Combine the two inequalities to get 1 < x < 5

## Example Question #8 : Inequalities

Starting with the first inequality:

Then, our second inequality tells us that

## Example Question #9 : Inequalities

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## Writing, Solving, and Graphing Inequalities - 6.EE.B.5 and 7.EE.B.4.b

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Summary · Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own. · But

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