Grade 6 Lessons

Multi Step Linear Inequalities

In this blog post, we will explore the concept of multi-step linear inequalities. We will define what they are and look at some examples to better understand how they work.

Definition of multi-step linear inequalities

Multi-step linear inequalities are mathematical expressions that involve multiple steps to solve for an unknown variable. They are similar to regular linear inequalities but require additional operations or steps to find the solution.

These inequalities often involve a combination of addition, subtraction, multiplication, and division. The goal is to isolate the variable on one side of the inequality sign to determine the range of values that satisfy the inequality.

Examples of multi-step linear inequalities

Let’s look at some examples to understand better how to solve multi-step linear inequalities.

Example 1:

Solve for x: 2x + 5 > 13

We need to isolate the variable x on one side to solve this inequality. First, we subtract 5 from both sides of the inequality:

2x + 5 – 5 > 13 – 52x > 8

Next, we divide both sides by 2 to solve for x:(2x)/2 > 8/2x > 4

So, the solution for this inequality is x > 4, which means any value greater than 4 satisfies the inequality.

Example 2:Solve for y: 3(y – 2) + 4 > 16 + 2y

To solve this inequality, we start by simplifying both sides of the equation:3y – 6 + 4 > 16 + 2y3y – 2 > 16 + 2y

Next, we subtract 2y from both sides to isolate the variable y:3y – 2 – 2y > 16 + 2y – 2yy – 2 > 16

Then, we add 2 to both sides to solve for y:y – 2 + 2 > 16 + 2y > 18

The solution for this inequality is y > 18, which means any value greater than 18 satisfies the inequality.

In conclusion, multi-step linear inequalities require multiple operations to solve for the unknown variable. They combine addition, subtraction, multiplication, and division to isolate the variable and determine the range of values satisfying the inequality. By practicing solving examples, you can improve your understanding and proficiency in working with multi-step linear inequalities.

Solving Multi-Step Linear Inequalities

The steps involved in solving multi-step linear inequalities

When it comes to solving multi-step linear inequalities, there are a few key steps that you need to follow:

Combine like terms on each side of the inequality.
Use inverse operations to isolate the variable on one side of the inequality.
Check whether the inequality sign must be flipped when performing an inverse operation.
Simplify both sides of the inequality until you have isolated the variable.
Express the solution using appropriate inequality symbols.

Solving inequalities with one variable

Suppose you’re dealing with an inequality with only one variable, such as x or y. In that case, you can follow the abovementioned steps to solve it. Let’s look at an example:

Example: Solve for x: 2x + 5 > 13

Subtract 5 from both sides of the inequality: 2x + 5 – 5 > 13 – 5
Divide both sides by 2 to solve for x: (2x)/2 > 8/2

Therefore, the solution to the inequality is x > 4. It means that any value greater than 4 satisfies the inequality.

Solving inequalities with multiple variables

Sometimes, you may encounter inequalities with multiple variables, such as x and y. Solving these types of inequalities is similar to solving those with one variable. Here’s an example:

Example: Solve for y: 3(y – 2) + 4 > 16 + 2y

Simplify both sides of the equation: 3y – 6 + 4 > 16 + 2y
Subtract 2y from both sides to isolate y: 3y – 2 – 2y > 16 + 2y – 2y

After simplifying, we get y > 18 as the solution to the inequality. It means that any value greater than 18 satisfies the inequality.

Following the steps and practicing examples can improve your understanding and proficiency in solving multi-step linear inequalities. Remember always to check your solution and double-check your work to ensure accuracy.

Graphing Multi-Step Linear Inequalities

Graphing linear inequalities on a number line

Graphing linear inequalities on a number line is a visual way to represent the solution to an inequality. It helps us understand the range of values that satisfy the inequality. Here’s how you can graph linear inequalities on a number line:

Determine the inequality symbol (>, <, ≥, ≤) and rewrite the inequality in slope-intercept form, if necessary.
Identify the slope and y-intercept of the linear equation, if applicable.
Plot the y-intercept on the number line.
Use the slope to determine whether to draw a solid or dashed line.
Draw the line in the appropriate direction to represent the inequality. Solid lines are used for “greater than or equal to” and “less than or equal to” inequalities. In contrast, dashed lines are used for “greater than” and “less than” inequalities.
Shade the region on the number line representing the inequality solution. We shade the region to the right of the line for “greater than” inequalities and to the left for “less than” inequalities.
Label the shaded region with the appropriate inequality symbol.

Graphing linear inequalities on a coordinate plane

Graphing linear inequalities on a coordinate plane can provide a visual representation of the solution to a system of inequalities. It allows us to see the overlapping regions of multiple inequalities. Here’s how you can graph linear inequalities on a coordinate plane:

Plot the inequality as an equation on the coordinate plane. If necessary, rearrange the inequality to be in slope-intercept form (y = mx + b).
Determine the slope and y-intercept of the linear equation.
Plot the first point on the graph using the y-intercept as a starting point.
Use the slope to find and plot a second point on the graph.
Draw a dashed or solid line that connects the two points. If the inequality includes “greater than or equal to” or “less than or equal to,” use a solid line. If the inequality includes “greater than” or “less than,” use a dashed line.
Shade the region that represents the solution to the inequality. Shade the area above the line for inequalities that involve “greater than” or “greater than or equal to.” Shade the region below the line for inequalities that involve “less than” or “less than or equal to.”
Repeat the process for each inequality in the system and identify the overlapping region where the shaded areas intersect.
Label the overlapping region with the appropriate inequality symbols.

By understanding and practicing graphing techniques for multi-step linear inequalities, you can effectively represent and solve these problems visually. Always double-check your work.

Solving Equations vs. Solving Inequalities

Difference between solving equations and solving inequalities

When it comes to solving equations and solving inequalities, there are a few key differences to keep in mind:

Equations have only one solution, which is the value that makes the equation true.
Inequalities have a range of values that can make the inequality true.
When solving equations, you use addition, subtraction, multiplication, and division operations to isolate the variable and find the solution.
When solving inequalities, you use similar operations to isolate the variable. Still, it would help if you also accounted for the inequality symbol and the direction of the inequality.
Inequalities can also have multiple solutions or no solutions at all, depending on the nature of the inequality.

Conclusion

Summary of multi-step linear inequalities

In conclusion, graphing multi-step linear inequalities can visually represent the solution to an inequality, whether it’s on a number line or a coordinate plane. You can accurately represent and solve these problems by following the steps outlined above. Additionally, it’s important to remember the critical differences between solving equations and solving inequalities, as they require different approaches and can have varying solutions. With practice and attention to detail, you can master graphing and solving multi-step linear inequalities.