Solving OneStep Linear Inequalities: A Comprehensive Guide Unlock the power of onestep linear inequalities with our easytofollow guide. Learn essential techniques, practice with realworld examples, and boost your algebra skills today.
 Solve.
 Determine if the solution is correct for each inequality.
 In a pizza restaurant, you need to order a minimum of $30 of pizzas to get a free drink. If each pizza is $7.45, how many pizzas do you need to order to get the free drink?
 Anna wants to save up $\frac{1}{5}$ of her total income each month. If she wants to save up $260 per month, how much money does she need to earn?
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Introduction to OneStep Linear Inequalities
Onestep linear inequalities are fundamental concepts in algebra that extend our understanding beyond equations. The introduction video provides a crucial foundation for grasping this topic. Unlike equations, which have a single solution, inequalities represent a range of values that satisfy a given condition. For example, while x = 5 is an equation with one solution, x > 5 is an inequality with multiple solutions. Inequalities use symbols like <, >, , or to show relationships between expressions. The solution set of an inequality includes all values that make the statement true. Understanding onestep linear inequalities is essential for solving inequalities and more complex problems in mathematics and realworld applications. By mastering this concept, students can develop critical thinking skills and a deeper appreciation for mathematical relationships. The introduction video serves as a gateway to exploring these ideas, setting the stage for more advanced topics in algebra and beyond.
Solve. $7x \le 63$
Step 1: Understand the Inequality
When solving onestep linear inequalities, it's important to understand that the process is similar to solving linear equations. The main difference is the inequality sign (, , <, >) instead of an equal sign (=). In this case, we have the inequality $7x \le 63$. Our goal is to isolate the variable $x$ on one side of the inequality.
Step 2: Treat the Inequality Like an Equation
To simplify the process, treat the inequality as if it were an equation. This means you can perform the same operations on both sides of the inequality as you would with an equation. For instance, if you had $7x = 63$, you would solve for $x$ by dividing both sides by 7. The same approach applies here.
Step 3: Divide Both Sides by 7
To isolate $x$, divide both sides of the inequality by 7. This operation will help you get $x$ by itself on one side of the inequality. Here's how it looks:
$7x \le 63$
Divide both sides by 7:
$\frac{7x}{7} \le \frac{63}{7}$
When you divide 7 by 7 on the left side, it cancels out, leaving you with $x$. On the right side, 63 divided by 7 equals 9. So, the inequality simplifies to:
$x \le 9$
Step 4: Interpret the Solution
The final inequality $x \le 9$ means that $x$ can be any number less than or equal to 9. This is the solution to the inequality. In other words, any value of $x$ that is 9 or smaller will satisfy the original inequality $7x \le 63$.
Step 5: Verify the Solution
To ensure the solution is correct, you can test a few values of $x$ that are less than or equal to 9. For example, if $x = 9$, substituting it back into the original inequality gives:
$7(9) \le 63$
$63 \le 63$
This is true. You can also test a value less than 9, such as $x = 5$:
$7(5) \le 63$
$35 \le 63$
This is also true. Therefore, the solution $x \le 9$ is verified.
Here are some frequently asked questions about solving onestep linear inequalities:

What is a onestep linear inequality?
A onestep linear inequality is a mathematical statement that compares two expressions using inequality symbols (<, >, , ) and can be solved in one step. For example, x + 3 > 7 is a onestep linear inequality.

How do you solve a onestep linear inequality?
To solve a onestep linear inequality, perform the inverse operation on both sides of the inequality to isolate the variable. For example, to solve x + 3 > 7, subtract 3 from both sides: x > 4.

What happens when you multiply or divide an inequality by a negative number?
When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. For example, if 2x < 6, dividing both sides by 2 gives x > 3.

How do you graph the solution of a onestep linear inequality on a number line?
To graph the solution, use an open circle () for strict inequalities (< or >) and a closed circle () for inclusive inequalities ( or ) at the boundary point. Then, shade the line in the direction that satisfies the inequality.

Can you give an example of a reallife application of onestep linear inequalities?
A reallife application could be determining how many hours you need to work to earn at least a certain amount of money. For instance, if you earn $15 per hour and want to make at least $300, you can set up the inequality 15x 300, where x is the number of hours worked.
Understanding the foundation of mathematics is crucial when tackling more advanced concepts like solving onestep linear inequalities. To excel in this area, it's essential to grasp several key prerequisite topics that form the building blocks of algebraic problemsolving.
One fundamental skill is solving equations, particularly those involving distance and time. This ability helps students develop a strong sense of algebraic reasoning and equation manipulation, which directly translates to working with inequalities. By mastering these types of problems, students can more easily transition to understanding the subtle differences between equations and inequalities.
While onestep linear inequalities are relatively straightforward, it's beneficial to have exposure to more complex concepts like graphing inequalities. This broader perspective allows students to visualize how inequalities work on a coordinate plane, reinforcing their understanding of the number line and inequality symbols used in simpler problems.
Additionally, familiarity with compound inequalities can provide valuable context. Although onestep inequalities are less complex, understanding how multiple inequalities can be combined helps students appreciate the versatility and importance of inequality concepts in mathematics.
A solid grasp of basic arithmetic operations is also crucial. Specifically, proficiency in multiplying by negative numbers and dividing by negative numbers is essential when solving inequalities. These operations often come into play when manipulating inequality expressions, and understanding how they affect the inequality sign is vital for arriving at correct solutions.
By mastering these prerequisite topics, students build a strong foundation for tackling onestep linear inequalities. The ability to solve equations provides the basic algebraic skills needed, while exposure to graphing and compound inequalities offers a broader context. Proficiency in working with negative numbers in multiplication and division ensures students can confidently manipulate inequality expressions.
As students progress through these prerequisite topics, they develop critical thinking skills and mathematical intuition that are invaluable when solving onestep linear inequalities. Each concept builds upon the last, creating a comprehensive understanding of algebraic relationships and inequalities. This solid foundation not only aids in solving onestep linear inequalities but also prepares students for more advanced mathematical concepts they will encounter in their future studies.
In conclusion, taking the time to thoroughly understand these prerequisite topics will greatly enhance a student's ability to solve onestep linear inequalities with confidence and accuracy. It's an investment in mathematical knowledge that pays dividends throughout one's academic journey and beyond.