Two Step Inequalities Worksheet, Examples, and Practice

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Key Points about Solving Two Step Inequalities

Two-step inequalities are mathematical expressions that involve two operations and represent a range of values that satisfy the given equation.
Solving two-step inequalities requires a solid understanding of basic algebraic concepts such as order of operations, inverse operations, and properties of inequality.
The process of solving two-step inequalities involves isolating the variable on one side of the inequality and determining the range of values that satisfy the equation.

What are Two Step Inequalities?

Two Step Inequalities are inequalities that take two steps to solve. This means that you have to add, subtract, multiply, or divide two times in order to solve the inequality. When solving each Two Step Inequality you will have to either add or subtract first and then multiply or divide second to solve the inequality. You can tell when an expression is an inequality by looking at the math symbol in the middle of the expression. If the expression has a greater than, less than, greater than or equal to, or less than or equal to symbol in the expression then the expression is an inequality. If you multiply or divide the inequality by a negative number, then the inequality sign must change directions.

Two-step inequalities are a fundamental concept in algebra that is essential for solving complex mathematical problems. A two-step inequality is a mathematical expression that involves two operations, such as addition and multiplication or subtraction and division. It is an inequality because it represents a range of values that satisfies the given equation.

Solving two-step inequalities requires a solid understanding of basic algebraic concepts such as order of operations, inverse operations, and properties of inequality. The process involves isolating the variable on one side of the inequality and determining the range of values that satisfy the equation. This process is similar to solving two-step equations, but with an added step of determining the direction of the inequality.

Common Core Standard: 7.EE.1
Related Topics: Combining Like Terms, Distributive Property, Two Step Equations, One Step Inequalities, Multi Step Inequalities
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Two Step Inequalities

How to Solve Two Step Inequalities

Two-step inequalities are slightly more complicated than one-step inequalities. However, they can be solved using a few simple steps. Here’s how to solve two-step inequalities:

Simplify the inequality: Start by simplifying the inequality as much as possible. This may involve combining like terms or distributing a negative sign. The goal is to get the inequality into the form of ax + b < c or ax + b > c.
Isolate the variable: Once the inequality is simplified, isolate the variable on one side of the inequality. This may involve adding or subtracting a constant from both sides of the inequality, or multiplying or dividing both sides of the inequality by a constant.
Check the solution: After isolating the variable, check the solution by plugging it back into the original inequality. If the solution makes the inequality true, then it is a valid solution. If not, then it is not a valid solution.

It’s important to note that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. For example, if you multiply both sides of the inequality -3x < 6 by -1, you get 3x > -6.

Here’s an example of how to solve a two-step inequality:

Solve 2x + 3 < 9.

Simplify the inequality: Subtract 3 from both sides of the inequality to get 2x < 6.
Isolate the variable: Divide both sides of the inequality by 2 to get x < 3.
Check the solution: Plug x = 3 back into the original inequality to get 2(3) + 3 < 9, which simplifies to 9 < 9. Since this is not true, x = 3 is not a valid solution. Therefore, the solution to the inequality is x < 3.

By following these simple steps, anyone can solve two-step inequalities with ease.

The Secret to Solving Two Step Inequalities

Isolation of Variable

When solving two-step inequalities, the first step is to isolate the variable on one side of the inequality sign. This involves performing inverse operations on both sides of the inequality to get the variable alone. The order of operations is the same as in algebraic equations, but with the added step of changing the inequality sign if a negative number is multiplied or divided.

Applying Inverse Operations

After isolating the variable, the next step is to apply inverse operations to solve for the variable. This involves undoing the operations that were performed on the variable to get it alone. For example, if the variable was multiplied by 3, it should be divided by 3 to undo the multiplication. If the variable was added to 5, it should be subtracted by 5 to undo the addition.

Checking the Solution

The final step in solving two-step inequalities is to check the solution. This involves plugging the solution back into the original inequality to see if it makes the inequality true. If the inequality is true, then the solution is correct. If the inequality is false, then the solution is incorrect.

It is important to note that the solution to a two-step inequality is not a single number, but rather a range of values that satisfy the inequality. This range can be represented graphically on a number line or in interval notation.

Overall, solving two-step inequalities requires a solid understanding of algebraic operations and the ability to apply inverse operations to isolate and solve for the variable. With practice, anyone can master the art of solving two-step inequalities and confidently tackle more complex algebraic problems.

Two Step Inequalities Solution

Solve Two Step Inequalities Examples in 3 Easy Steps

Two-step inequalities are a bit more complicated than one-step inequalities, but they follow the same basic principles.

Add or subtract the constant to get just the variable on one side of the inequality.
Multiply or divide so that you cancel out the coefficient on the term.
If you have to multiply or divide by a negative number then the inequality symbol changes directions.

Here are some examples of two-step inequalities:

Example 1

Solve for x: 3x + 5 < 14

To solve this inequality, you need to isolate x on one side of the inequality sign.

Subtract 5 from both sides: 3x < 9
Divide both sides by 3: x < 3

Therefore, the solution to the inequality is x < 3.

Example 2

Solve for y: 2y + 7 > 15

To solve this inequality, you need to isolate y on one side of the inequality sign.

Subtract 7 from both sides: 2y > 8
Divide both sides by 2: y > 4

Therefore, the solution to the inequality is y > 4.

Example 3

Solve for z: -4z + 3 < 11

To solve this inequality, you need to isolate z on one side of the inequality sign.

Subtract 3 from both sides: -4z < 8
Divide both sides by -4 (note that dividing by a negative number flips the inequality sign): z > -2

Therefore, the solution to the inequality is z > -2.

Example 4

Solve for a: 2a – 5 > 7a + 1

To solve this inequality, you need to isolate a on one side of the inequality sign.

Subtract 2a from both sides: -5 > 5a + 1
Subtract 1 from both sides: -6 > 5a
Divide both sides by 5: a < -6/5

Therefore, the solution to the inequality is a < -6/5.

These examples demonstrate how to solve two-step inequalities by applying inverse operations to isolate the variable on one side of the inequality sign.

5 Quick Two Step Inequalities Practice Problems

Two Step Inequalities Word Problems

Two-step inequalities word problems are a common type of math problem that involve solving inequalities with two steps. These problems require students to use their math skills to interpret and solve real-world situations.

To solve a two-step inequality word problem, students need to identify the variable, write an inequality equation, and then solve it. The solution to the inequality equation is the range of values that satisfy the problem’s conditions.

For instance, a common two-step inequality word problem involves finding the range of possible values for a variable. For example, “Katie wants to collect over 100 seashells. She already has 35 seashells. How many more seashells does she need to collect?”

To solve this problem, students can start by identifying the variable, which is the number of seashells Katie needs to collect. Let’s call this variable “x”.

Next, students can write an inequality equation that represents the problem’s conditions. In this case, the inequality equation would be “x + 35 > 100”, which means that the number of seashells Katie needs to collect plus the number she already has is greater than 100.

Finally, students can solve the inequality equation to find the range of possible values for “x”. In this case, the solution would be “x > 65”, which means that Katie needs to collect more than 65 seashells to reach her goal.

Other two-step inequality word problems may involve finding the maximum or minimum value of a variable, or finding the range of values that satisfy multiple conditions. These problems require students to use their math skills to interpret the problem’s conditions and write an appropriate inequality equation.

Overall, two-step inequality word problems are an important part of math education that help students develop critical thinking and problem-solving skills. By practicing these problems, students can improve their math skills and prepare for more advanced math concepts.

Common Mistakes in Solving Two Step Inequalities

When solving two-step inequalities, there are some common mistakes that students often make. Here are some of the most frequent mistakes and how to avoid them:

Mistake 1: Forgetting to Flip the Inequality

When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be flipped. This is a common mistake that can lead to incorrect solutions. For example, if solving the inequality 2x – 5 > 7, adding 5 to both sides gives 2x > 12. Dividing both sides by -2 gives x < -6, but if the inequality sign is not flipped, the solution will be x > -6, which is incorrect. To avoid this mistake, always remember to flip the inequality sign when multiplying or dividing by a negative number.

Mistake 2: Combining Like Terms Incorrectly

When solving two-step inequalities, it is important to combine like terms correctly. This is another common mistake that can lead to incorrect solutions. For example, if solving the inequality 3x + 2 > 11, subtracting 2 from both sides gives 3x > 9. Dividing both sides by 3 gives x > 3. However, if the 2 is mistakenly subtracted from the 3x instead of the constant term, the solution will be x > 3.33, which is incorrect. To avoid this mistake, always combine like terms correctly before solving the inequality.

Mistake 3: Forgetting to Check the Solution

When solving two-step inequalities, it is important to check the solution to make sure it satisfies the inequality. This is a common mistake that can lead to incorrect solutions. For example, if solving the inequality 2x + 3 < 7, subtracting 3 from both sides gives 2x < 4. Dividing both sides by 2 gives x < 2. However, if the solution is not checked, it may be missed that x = 2 also satisfies the inequality. To avoid this mistake, always check the solution by plugging it back into the original inequality to make sure it satisfies the inequality.

By avoiding these common mistakes, students can solve two-step inequalities accurately and confidently.

How to do Two Step Inequalities FAQ

What is a two-step inequality?

A two-step inequality is a type of inequality that requires two steps to solve. It contains two operations, such as addition, subtraction, multiplication, or division, which must be performed in a specific order to solve the inequality. Two-step inequalities are often used in math problems that involve finding a range of possible values for a variable.

How do you solve a two-step inequality?

To solve a two-step inequality, you must first isolate the variable on one side of the inequality sign. This is done by performing the inverse operation of the operation that is furthest away from the variable. Then, you can perform the inverse operation of the operation that is closest to the variable to isolate the variable completely. If you multiply or divide by a negative number, the inequality sign must be reversed.

What is a two-step equation?

A two-step equation is a type of equation that requires two steps to solve. It contains two operations, such as addition, subtraction, multiplication, or division, which must be performed in a specific order to solve the equation. Two-step equations are often used in math problems that involve finding the value of a variable.

How do you solve one and two-step inequalities?

To solve one and two-step inequalities, you must first isolate the variable on one side of the inequality sign. This is done by performing the inverse operation of the operation that is furthest away from the variable. Then, you can perform the inverse operation of the operation that is closest to the variable to isolate the variable completely. If you multiply or divide by a negative number, the inequality sign must be reversed.

How do you solve a two-step inequality word problem?

To solve a two-step inequality word problem, you must first identify the inequality and the variable. Then, you must translate the words into an inequality and solve it using the same steps as a regular two-step inequality. It is important to pay attention to the wording of the problem to ensure that you are translating it correctly.

How to do two-step inequalities 7th grade?

In 7th grade, students learn how to solve two-step inequalities using the same steps as for one-step inequalities. They also learn how to translate word problems into inequalities and solve them using the same steps. It is important for students to practice solving different types of inequalities to build their skills and confidence.

What is the two-step inequalities formula?

There is no specific formula for solving two-step inequalities. Instead, you must use the same steps as for one-step inequalities, but with an additional operation. It is important to remember to perform the inverse operations in the correct order and to reverse the inequality sign if necessary.

Two Step Inequalities Worksheet Video Explanation

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

Watch the free Two Step Inequalities video on YouTube here: Two Step Inequalities Video

Video Transcript:
This video is about solving 2 step inequalities. You can get the two-step inequalities worksheet with answers used in this video for free by clicking on the link in the description below. In order to solve two step inequalities, you follow rules that are very similar to solving two step equations. The main difference being that instead of an equal sign in the middle of your equation, or inequality, you have one of four types of inequality symbols.

The first thing to solving inequalities is that you solve it just like you would for an equation and that you solve for the variable. You’re still going to try to isolate the variable on one side of the inequality. The second thing you must remember about inequalities and what makes it a little bit different than solving for equations is that if you multiply or divide by a negative you have to flip the sign of your inequality symbol. The only time you do this is when you multiply or divide by a negative number.

Now we’re going to do a real quick example of solving a two step inequality. In this example we have 3x minus 25 is less than 35. We’re going to draw our line down the middle of our inequality. We know that we have to isolate this x by getting it on one side of the inequality by itself. The first thing we need to do is we need to get rid of this minus 25. We’re going to add 25 to both sides. This 25 will cancel.

We bring down the 3x, our less than sign in the middle, and then 35 plus 25 is 60. Then this is like saying 3 times x the opposite of 3 times something is to divide by 3. we’re going to divide both sides by 3 and on this side we have x. Then we have our less than sign in the middle and then 60 divided by 3 is 20. You can see this is very similar to solving for equations except we have an inequality sign in the middle of our inequality. And if you divide by a negative or multiply by a negative number you will flip that sign in the middle. Let’s do a couple practice problems on our two step inequalities worksheet.

The first thing we’re going to do on our two step inequalities practice worksheet is number one. This problem gives us 9x minus 9 is greater than 9. We know we have to isolate the x and get the x by itself on one side of the inequality. In order to do that we have this minus 9 which is the first thing we have to get rid of. We’re going to add 9 to both sides because the opposite of minus 9 is plus 9.

These will cancel on the left side and we have 9x. Then in the middle we still have greater than and then 9 plus 9 is 18. Now 9x is greater than 18 is what our inequality states. This is like saying 9 times x and the opposite of 9 times x is to divide by 9. That will cancel whatever you do to this side you also have to do this side. We’re going to divide by 9 on this side. Now we have x on the left the greater than symbol in the middle and then 18 divided by 9 on the right which is equal to 2. Our final solution is x is greater than 2.

Jumping down to number three on our two step inequalities worksheet we have negative 10x plus 14 is greater than or equal to 134. Again we’re going to follow the same steps that we just did or like just like you do when you’re solving for two step equations. The first thing we need to do is get rid of this plus 14, so we’re trying to get this x by itself. We have to get rid of this plus 14 so we subtract 14 here so that it cancels and then whatever you do on one side you also have to do the other. We subtract 14 here on this side.

We have negative 10x is greater than or equal to in the middle and then 134 minus 14 is 120. Then the next step is to get rid of this negative 10 that’s being multiplied times the x. The opposite of that is to divide by negative 10 so that the tens cancel and the negatives cancel and then whatever you do to one side you do the other. We divide this side by negative 10. Now what is different about inequalities is that because we just divided by a negative number, we have to flip the sign in the middle. This greater than or equal to turns into a less than or equal to symbol, so it switches. It’s different it’s changed then you just solve. 120 divided by negative 1 is negative 12. Our answer is x is less than or equal to negative 12.

The last problem we’re going to do on our two step inequalities worksheet is number seven. This problem gives us negative x minus eight is less than 31. Again we’re going to draw our line down the middle to get this x by itself on one side. We’re going to add 8 so that the 8 will cancel. Whatever due to one side you do the other.

We bring down our negative x our symbol in the middle, is less than, and then 31 plus 8 is 39. Now this negative x is like saying negative 1, even though it’s not written, times x. We have to get rid of this negative 1 by dividing by negative 1. We divide this side by negative 1. Now we have x and remember we divided by a negative so the symbol in the middle has to change. Our less than symbol becomes a greater than symbol and then 39 divided by negative 1 is negative 39 and that’s our solution. Hopefully you found this article and video helpful when learning how to solve 2 step inequalities.