# Multi Step Inequalities Worksheet, Examples, Practice

Get the free Multi Step Inequalities worksheet and other resources for teaching & understanding Multi Step Inequalities

### Key Points about Multi Step Inequalities

- Multi-step inequalities require more than one operation to solve.
- Solving multi-step inequalities involves using a combination of algebraic techniques, such as distributing, combining like terms, and isolating variables.
- Maintaining the balance of the inequality is crucial, and it is important to perform the same operation on both sides.

## What are Multi Step Inequalities?

Multi-step inequalities are a common topic in algebra courses, and understanding how to solve them is essential for success in higher-level math courses. Multi-step inequalities are inequalities that require more than one operation to solve. They are often used to model real-world problems, such as determining the minimum and maximum values of a function subject to certain constraints.

Solving multi-step inequalities involves using a combination of algebraic techniques, such as distributing, combining like terms, and isolating variables. Inequalities with variables on both sides require additional steps, such as combining like terms and then isolating the variable. Multi-step inequalities can also involve parentheses, which require distributing before combining like terms. As with solving equations, it is important to maintain the balance of the inequality by performing the same operation on both sides.

Solving Multi Step Inequalities is when a linear inequality has multiple Like Terms on the same side of the inequality symbol. The first thing you must do is realize that Like Terms exist on the same side of the inequality symbol. Once you have recognized that the inequality has Like Terms on the same side, you combine them by either adding them together or subtracting them.

After you Combine the Like Terms, you solve the inequality just like any other Two Step Inequality. The first step in solving Two Step Inequalities is to get all of the constants (numbers) on one side of the inequality symbol, and the coefficient with the variable on the other side. In order to do this you must use the addition and subtraction property of equality to get the constants on the opposite side as the variable. Once the constants are separated from the variable, you must use the multiplication or division property of equality to cancel out the coefficient on the variable.

You can always check your answer by substituting your solution back in to the inequality for the variable. If you multiply or divide by a negative number when solving your Multi Step Inequality you must change the direction of the inequality symbol.

**Common Core Standard: **7.EE.1**Related Topics:**Combining Like Terms, Distributive Property, Two Step Equations, One Step Inequalities, Two Step Inequalities**Return To: **Home, 7th Grade

## How to Solve Multi Step Inequalities

Multi-step inequalities are inequalities that require more than one step to solve. They involve two or more operations such as addition, subtraction, multiplication, and division. Solving multi-step inequalities can be challenging, but it is crucial to understand the process to solve them correctly.

### Importance of Order of Operations

The order of operations is essential when solving multi-step inequalities. The order of operations is a set of rules that dictate the order in which operations are performed in an equation or inequality. The order of operations is as follows:

- Parentheses and other grouping symbols
- Exponents and roots
- Multiplication and division (performed from left to right)
- Addition and subtraction (performed from left to right)

It is crucial to follow the order of operations when solving multi-step inequalities. Failing to do so can result in an incorrect solution.

For example, consider the inequality 2x + 3 > 7x – 5. To solve this inequality, one must first simplify both sides by combining like terms.

2x + 3 > 7x – 5

Subtracting 2x from both sides, we get:

3 > 5x – 5

Adding 5 to both sides, we get:

8 > 5x

Dividing both sides by 5, we get:

1.6 > x

Therefore, the solution to the inequality 2x + 3 > 7x – 5 is x < 1.6.

In conclusion, solving multi-step inequalities requires following the order of operations and simplifying the inequality step by step.

## Solving Inequalities with Variables on Both Sides

When solving multi-step inequalities, one may encounter inequalities with variables on both sides. These types of inequalities can be solved by using the same principles used for solving equations with variables on both sides. The goal is to isolate the variable on one side of the inequality sign and obtain the solution set.

To solve an inequality with variables on both sides, one should follow these steps:

- Simplify both sides of the inequality by combining like terms.
- Move the variable terms to one side of the inequality sign and the constant terms to the other side.
- Use the properties of inequality to maintain the inequality sign. For example, when multiplying or dividing by a negative number, the inequality sign should be flipped.
- Simplify the resulting inequality and obtain the solution set.

It is important to note that when solving inequalities with variables on both sides, one should be careful not to combine unlike terms or distribute incorrectly. This can lead to errors in the solution set.

Let’s take a look at an example:

**Example:** Solve the inequality 3x – 4 > 2x + 5.

**Solution:**

- Simplify both sides by combining like terms: 3x – 2x > 5 + 4
- Move the variable terms to one side and the constant terms to the other: x > 9
- Since there is no multiplication or division by a negative number, the inequality sign remains the same.
- The solution set is x > 9.

In conclusion, solving inequalities with variables on both sides requires the same principles used for solving equations with variables on both sides. One should simplify both sides, move variable terms to one side and constant terms to the other, use the properties of inequality, and simplify to obtain the solution set.

## Multi Step Inequalities Definition

Multi-step inequalities are mathematical expressions that involve more than one inequality statement and require multiple steps to solve. They are used to represent relationships between variables that are not equal, but rather greater or less than each other. Multi-step inequalities are commonly used in algebraic equations to solve real-world problems, such as calculating the minimum or maximum value of a function.

To solve a multi-step inequality, one must use a series of algebraic operations to isolate the variable on one side of the inequality and determine the range of values that satisfy the inequality. The same rules of algebra that apply to equations also apply to inequalities, with the exception that the direction of the inequality sign must be reversed when multiplying or dividing by a negative number.

Multi-step inequalities can be represented in a variety of ways, including linear and quadratic equations, absolute value equations, and systems of equations. They are often used in fields such as finance, engineering, and science to model complex relationships between variables and make predictions about future outcomes.

Overall, multi-step inequalities are an important concept in mathematics that play a crucial role in solving a wide range of real-world problems. By understanding the definition and properties of multi-step inequalities, one can develop the skills necessary to solve complex algebraic equations and make informed decisions based on mathematical models and data analysis.

## Multi Step Inequalities Examples: Solve in 4 Easy Steps

- Combine Like terms that are on the same side of the inequality by adding them together or subtracting them from each other.
- Add or subtract the constants so that you get the variable and constant on opposite side of the inequality.
- Divide the entire inequality by the coefficient on the variable to solve the inequality.
- If you multiplied or divided by a negative number you have to change the direction of the inequality symbol.

Multi-step inequalities are inequalities that require more than one step to solve. They involve combining multiple operations such as addition, subtraction, multiplication, and division to isolate the variable. Here are some examples of multi-step inequalities:

### Example 1:

Solve for x: 2(x+3) – 5 > x + 10

First, distribute the 2:

2x + 6 – 5 > x + 10

Combine like terms:

x + 1 > x + 10

Subtract x from both sides:

1 > 10

This is a contradiction, so there is no solution.

### Example 2:

Solve for y: 3(y-2) + 4 > 2y + 1

First, distribute the 3:

3y – 6 + 4 > 2y + 1

Combine like terms:

3y – 2 > 2y + 1

Subtract 2y from both sides:

y – 2 > 1

Add 2 to both sides:

y > 3

The solution is y > 3.

### Example 3:

Solve for z: 5(z+2) – 3(z-1) < 12

First, distribute the 5 and -3:

5z + 10 – 3z + 3 < 12

Combine like terms:

2z + 13 < 12

Subtract 13 from both sides:

2z < -1

Divide by 2:

z < -1/2

The solution is z < -1/2.

These examples demonstrate the process of solving multi-step inequalities. It is important to carefully follow each step and check the solution by substituting it back into the original inequality.

## 5 Quick Multi Step Inequalities Practice Problems

## Solving Multi Step Inequalities with Addition and Subtraction

Multi-step inequalities are solved just like multi-step equations except that you have to be aware of when to reverse the inequality symbol. The symbol should be reversed if:

- You reverse the sides of the equation.
- You multiply or divide both sides by a negative number.

### Step by Step Approach

To solve multi-step inequalities with addition and subtraction, follow these steps:

- Simplify both sides of the inequality by combining like terms.
- Isolate the variable term by moving the constant terms to the other side of the inequality.
- Simplify the inequality by combining like terms.
- Divide both sides of the inequality by the coefficient of the variable term if necessary.

Here is an example:

```
2x + 5 > 11
```

- Simplify both sides of the inequality by combining like terms.

```
2x > 6
```

- Isolate the variable term by moving the constant terms to the other side of the inequality.

```
2x - 5 > 6 - 5
2x > 1
```

- Simplify the inequality by combining like terms.

```
x > 1/2
```

- Divide both sides of the inequality by the coefficient of the variable term if necessary.

### Common Mistakes

When solving multi-step inequalities with addition and subtraction, there are a few common mistakes that students make. These include:

- Forgetting to reverse the inequality symbol when multiplying or dividing both sides of the inequality by a negative number.
- Combining terms incorrectly, especially when dealing with negative numbers.
- Not isolating the variable term correctly, which can lead to incorrect solutions.

To avoid these mistakes, it is important to double-check your work and take your time when solving multi-step inequalities.

## Solving Multi Step Inequalities with Multiplication and Division

Multi-step inequalities are a bit more complicated than one-step inequalities. However, the process of solving multi-step inequalities with multiplication and division can be broken down into a simple step-by-step approach.

### Step by Step Approach

To solve multi-step inequalities with multiplication and division, follow these steps:

- Simplify both sides of the inequality by combining like terms.
- Isolate the variable term on one side of the inequality by adding or subtracting constants.
- Multiply or divide both sides of the inequality by a positive number to isolate the variable.
- Check the solution by substituting it back into the original inequality.

For example, let’s solve the inequality 4x + 3 > 11x – 7:

- Simplify both sides of the inequality by combining like terms: 3 + 7 = 10, so the inequality becomes 4x + 10 > 11x.
- Isolate the variable term on one side of the inequality by subtracting 4x from both sides: 10 > 7x.
- Divide both sides of the inequality by 7 to isolate the variable: 10/7 > x.
- Check the solution: Substitute x = 10/7 back into the original inequality: 4(10/7) + 3 > 11(10/7) – 7. Simplifying both sides, we get 19/7 > 19/7, which is true.

### Common Mistakes

When solving multi-step inequalities with multiplication and division, there are some common mistakes to avoid:

- Forgetting to combine like terms before isolating the variable.
- Dividing by a negative number, which requires reversing the direction of the inequality.
- Forgetting to check the solution by substituting it back into the original inequality.

By following the step-by-step approach and avoiding common mistakes, anyone can solve multi-step inequalities with multiplication and division with confidence and accuracy.

## Solving Multi Step Inequalities FAQ

### How do you solve inequalities with fractions?

To solve inequalities with fractions, the first step is to eliminate the denominators by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators. After that, the inequality can be solved using the same methods as for equations.

### What are compound inequalities?

Compound inequalities are inequalities that contain two or more inequalities connected by the words “and” or “or”. “And” compound inequalities require both inequalities to be true, while “or” compound inequalities require at least one of the inequalities to be true.

### What is the process for solving multi-step inequalities with variables on both sides?

The process for solving multi-step inequalities with variables on both sides is similar to the process for solving multi-step equations. The goal is to isolate the variable on one side of the inequality by applying inverse operations in the correct order. It’s important to remember to reverse the direction of the inequality if multiplying or dividing by a negative number.

### What are the steps to solving multi-step equations?

The steps to solving multi-step equations are to simplify both sides of the equation, isolate the variable by applying inverse operations in the correct order, and check the solution by plugging it back into the original equation.

### What are the steps to solving an inequality?

The steps to solving an inequality depend on the type of inequality. For one-step inequalities, simply isolate the variable on one side of the inequality. For multi-step inequalities, simplify both sides of the inequality, isolate the variable by applying inverse operations in the correct order, and remember to reverse the direction of the inequality if multiplying or dividing by a negative number.

### How can you solve inequalities with two variables?

To solve inequalities with two variables, graph the inequality on the coordinate plane and shade the region that satisfies the inequality. The solution is the set of all points in the shaded region.

### What is a multi-step inequality?

A multi-step inequality is an inequality that requires more than one step to solve. It typically involves combining multiple inequalities or applying inverse operations in the correct order to isolate the variable.

### What is the order of solving inequalities?

The order of solving inequalities is to simplify both sides of the inequality, isolate the variable by applying inverse operations in the correct order, and remember to reverse the direction of the inequality if multiplying or dividing by a negative number.

## Multi Step Inequalities Worksheet Video Explanation

Watch our free video on how to solve Multi Step Inequalities. This video shows how to solve problems that are on our free solving Multi Step Inequalities worksheet that you can get by submitting your email above.

**Watch the free Multi Step Inequalities video on YouTube here: Multi Step Inequalities Video**

**Video Transcript:**

This video is about solving multi-step inequalities. You can get the multi-step inequalities worksheet used in this video for free by clicking on the link in the description below. Solving any multi step inequalities examples is very similar when solving for multi step equations. You will combine like terms, you’ll solve for the variable, and then if you divide by a negative number it’s going to change the direction of the symbol of your inequality.

This first example problem gives us negative 3x minus 5 plus 6x is less than or equal to 46. The first step in this multi step inequality is to combine like terms. If you combine negative 3x plus 6x you get positive 3x. You bring down your minus 5, bring down your less than or equal to symbol, and the 46. The next step is to get x by itself. We’re going to add 5 to this side and to this side.

The 5’s cancel and we’re left with 3x is less than or equal to 51. Then the final step is to divide both sides by 3 so that you get x by itself on this side. These threes will cancel and then 51 divided by 3 is 17. The solution is x is less than or equal to 17. Let’s do a couple practice problems on our multi step inequalities worksheet.

The first problem we’re going to do on our multistep inequalities worksheet is number one. The problem gives us 1 plus 4x plus 3 is greater than 20. The first thing we need to do is we need to combine like terms. In this case we’re going to add 1 plus 3 because they are the like terms on the left side of the inequality and 1 plus 3 is 4. We bring down the 4x and then we combine 1 and 3. We bring down our greater than symbol and then 20.

Then the next step is we have to get x by itself. We’re going to subtract 4 from both sides so that this 4 cancels and we’re left with 4x is greater than 20 minus 4 is 16. The last step is we have to get rid of this 4x. This is like saying 4 times x. The opposite of that is to divide by 4. These fours cancel and we’re left with x is greater than 16, divided by four is four. The solution is x is greater than four.

The next problem we’re going to do on the solve multi step inequalities worksheet is number six. This problem gives us 10x plus three plus 4x is greater than or equal to negative 25. Again, the first step is to combine like terms. We combine 10x plus 4x and we get 14x. Bring down the plus 3 and the greater than symbol and the negative 25. Then we have to get x on one side by itself.

We’re going to subtract 3 from this side so that the 3s cancel, whatever you do to one side you also have to do the other. We subtract 3 on this side and we bring down our 14x is greater than or equal to negative 25. Minus 3 is negative 28. And then 14 times x we have to undo. We divide both sides by 14. The 14s cancel we’re left with just x on this side is greater than or equal to negative 28 divided by 14 is negative two. Our answer is x is greater than or equal to negative two.

Finally, the last problem on our multi step inequalities worksheet is going to be number seven. We have negative x plus one minus x is less than or equal to negative 77. Again, the first step is to combine like terms. We have negative x and this is like negative 1x minus another negative 1x. Negative 1 minus 1 is negative 2. We have negative 2x, bring down our plus 1, our less than or equal to symbol, and our negative 77.

Then we have to get x on one side by itself. We’re going to subtract 1 here so that the ones cancel. We do minus 1 here so now we have negative 2x is less than or equal to negative 77 minus 1 is negative 78. Then we have to undo this negative 2x. This is like negative 2 times x so we divide by negative 2 so that the negatives and the 2s cancel. Whatever you do to one side you do the other. You divide this side by negative 2 as well.

We have x on this side, now we divided by a negative number and because we divided by a negative it’s going to change our symbol from less than or equal to, to greater than or equal to. Now we have x is greater than or equal to and then negative 78 divided by negative 2 is positive 39. The solution is x is greater than or equal to 39.

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