Not distributing the coefficient correctly

Oct 13, 2025 by recoveryshake.com 24 views

Introduction

Inequalities are mathematical expressions that compare two values, often using greater than, less than, greater than or equal to, or less than or equal to symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving the inequality $${ 2(3-x) \leq 5-4x }$, which is a fundamental concept in algebra and mathematics.

Understanding the Inequality

The given inequality is $${ 2(3-x) \leq 5-4x }$. To begin solving this inequality, we need to simplify the left-hand side by distributing the 2 to the terms inside the parentheses. This gives us $${ 6-2x \leq 5-4x }$. Now, we can combine like terms and isolate the variable x.

Distributing and Simplifying

To distribute the 2 to the terms inside the parentheses, we multiply 2 by each term inside the parentheses. This gives us $${ 6-2x \leq 5-4x }$. Now, we can combine like terms by adding or subtracting the same value to both sides of the inequality. In this case, we can add 2x to both sides to get $${ 6 \leq 5-2x }$.

Isolating the Variable

Now that we have simplified the inequality, we can isolate the variable x by adding 2x to both sides. This gives us $${ 6+2x \leq 5 }$. Next, we can subtract 6 from both sides to get $${ 2x \leq -1 }$. Finally, we can divide both sides by 2 to get $${ x \leq -\frac{1}{2} }$.

Checking the Solution

To check our solution, we can plug in a value of x that satisfies the inequality. Let's choose x = -1, which is less than or equal to -1/2. Plugging this value into the original inequality, we get $${ 2(3-(-1)) \leq 5-4(-1) }$, which simplifies to $${ 2(4) \leq 5+4 }$, or $${ 8 \leq 9 }$. Since this is true, our solution is correct.

Conclusion

Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we solved the inequality $${ 2(3-x) \leq 5-4x }$ by distributing and simplifying the left-hand side, combining like terms, and isolating the variable x. We also checked our solution by plugging in a value of x that satisfies the inequality. By following these steps, you can solve any inequality that comes your way.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid. Here are a few:

Not distributing the coefficient correctly: When distributing a coefficient to terms inside parentheses, make sure to multiply the coefficient by each term.
Not combining like terms correctly: When combining like terms, make sure to add or subtract the same value to both sides of the inequality.
Not isolating the variable correctly: When isolating the variable, make sure to add or subtract the same value to both sides of the inequality.
Not checking the solution: When checking the solution, make sure to plug in a value of x that satisfies the inequality.

Real-World Applications

Inequalities have many real-world applications, including:

Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Computer Science: Inequalities are used to solve problems in computer science, such as sorting and searching algorithms.

Final Thoughts

Solving inequalities is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve any inequality that comes your way. Remember to distribute and simplify the left-hand side, combine like terms, and isolate the variable x. Also, be sure to check your solution by plugging in a value of x that satisfies the inequality. With practice and patience, you will become proficient in solving inequalities and be able to apply them to real-world problems.

Introduction

In our previous article, we discussed how to solve the inequality $${ 2(3-x) \leq 5-4x }$. In this article, we will answer some common questions that students often have when solving inequalities.

Q: What is the difference between an inequality and an equation?

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that says one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.

Q: How do I know which direction to go when solving an inequality?

A: When solving an inequality, you need to maintain the direction of the inequality. If the inequality is greater than or equal to, you need to add or subtract the same value to both sides. If the inequality is less than or equal to, you need to subtract or add the same value to both sides.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form ax + b ≤ c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c ≤ d, where a, b, c, and d are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign chart method to determine the solution set.

Q: What is the sign chart method?

A: The sign chart method is a technique used to determine the solution set of a quadratic inequality. You create a chart with the x-values on one axis and the corresponding values of the quadratic expression on the other axis. Then, you shade in the regions where the quadratic expression is positive or negative.

Q: How do I know when to use the sign chart method?

A: You should use the sign chart method when the quadratic expression cannot be factored easily.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that involves a rational expression, while a polynomial inequality is an inequality that involves a polynomial expression.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the zeros of the numerator and denominator, and then use the sign chart method to determine the solution set.

Q: What is the difference between a linear inequality and a nonlinear inequality?

A: A linear inequality is an inequality that can be written in the form ax + b ≤ c, where a, b, and c are constants. A nonlinear inequality is an inequality that cannot be written in this form.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you need to use a variety of techniques, such as graphing, factoring, and the sign chart method.

Q: What is the difference between a one-sided inequality and a two-sided inequality?

A: A one-sided inequality is an inequality that has only one direction, such as x ≤ 2. A two-sided inequality is an inequality that has two directions, such as x ≤ 2 and x ≥ -3.

Q: How do I solve a one-sided inequality?

A: To solve a one-sided inequality, you need to add or subtract the same value to both sides of the inequality.

Q: What is the difference between a strict inequality and a nonstrict inequality?

A: A strict inequality is an inequality that has a strict inequality sign, such as x < 2. A nonstrict inequality is an inequality that has a nonstrict inequality sign, such as x ≤ 2.

Q: How do I solve a strict inequality?

A: To solve a strict inequality, you need to add or subtract the same value to both sides of the inequality.

Q: What is the difference between a linear inequality and a nonlinear inequality in three variables?

A: A linear inequality in three variables is an inequality that can be written in the form ax + by + cz ≤ d, where a, b, c, and d are constants. A nonlinear inequality in three variables is an inequality that cannot be written in this form.

Q: How do I solve a linear inequality in three variables?

A: To solve a linear inequality in three variables, you need to use a variety of techniques, such as graphing, factoring, and the sign chart method.

Q: What is the difference between a linear inequality and a nonlinear inequality in four variables?

A: A linear inequality in four variables is an inequality that can be written in the form ax + by + cz + dw ≤ e, where a, b, c, d, and e are constants. A nonlinear inequality in four variables is an inequality that cannot be written in this form.

Q: How do I solve a linear inequality in four variables?

A: To solve a linear inequality in four variables, you need to use a variety of techniques, such as graphing, factoring, and the sign chart method.

Conclusion

Solving inequalities is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve any inequality that comes your way. Remember to maintain the direction of the inequality, use the sign chart method when necessary, and be careful when working with rational and nonlinear inequalities. With practice and patience, you will become proficient in solving inequalities and be able to apply them to real-world problems.