Solve The Inequality $(35x - 14) \\\\leq \\\\frac{21x}{2} + 3$ And Graph The Solution On A Number Line.

In this comprehensive guide, we will delve into the process of solving and graphing the inequality $(35x - 14) \\leq \frac{21x}{2} + 3$. This type of problem is a fundamental concept in algebra, often encountered in various mathematical and real-world scenarios. We will break down the solution step-by-step, ensuring a clear understanding of each stage. Additionally, we will visually represent the solution set on a number line, a crucial skill for interpreting inequalities.

Understanding Linear Inequalities

Before we tackle the specific inequality, let's grasp the basics of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), $\\\\\leq$ (less than or equal to), and $\\\\\geq$ (greater than or equal to). Solving an inequality involves finding the range of values that satisfy the given condition. Unlike equations that have specific solutions, inequalities often have a range of solutions, which can be visually represented on a number line.

The inequality we are addressing, $(35x - 14) \\\\leq \\\\frac{21x}{2} + 3$, is a linear inequality because the variable 'x' is raised to the power of 1. The goal is to isolate 'x' on one side of the inequality to determine the solution set. This process involves applying algebraic operations while adhering to certain rules, particularly when multiplying or dividing by a negative number.

Step-by-Step Solution

1. Eliminate the Fraction

Our first step in solving the inequality $(35x - 14) \\\\leq \\\\frac{21x}{2} + 3$ is to eliminate the fraction. This makes the equation easier to manipulate and solve. To do this, we multiply both sides of the inequality by the denominator of the fraction, which in this case is 2. This ensures that we maintain the balance of the inequality while removing the fractional term.

Multiplying both sides by 2, we get:

$2(35x - 14) \\\\leq 2(\\\\frac{21x}{2} + 3)$

Distributing the 2 on both sides, we have:

$70x - 28 \\\\leq 21x + 6$

This step simplifies the inequality significantly, making it easier to proceed with isolating the variable 'x'. By eliminating the fraction, we've transformed the inequality into a more manageable form, setting the stage for the subsequent steps in solving for 'x'.

2. Isolate the Variable Terms

Now that we've eliminated the fraction, the next crucial step is to isolate the terms containing the variable 'x' on one side of the inequality. In the inequality $70x - 28 \\\\leq 21x + 6$, we achieve this by subtracting $21x$ from both sides. This operation ensures that all 'x' terms are grouped together, simplifying the equation further.

Subtracting $21x$ from both sides, we get:

$70x - 21x - 28 \\\\leq 21x - 21x + 6$

This simplifies to:

$49x - 28 \\\\leq 6$

By performing this step, we've successfully moved all the variable terms to the left side of the inequality, making it easier to isolate 'x' in the following steps. This is a standard algebraic technique used to solve equations and inequalities, and it's essential for arriving at the solution.

3. Isolate the Constant Terms

After isolating the variable terms, our next goal is to isolate the constant terms on the opposite side of the inequality. In the inequality $49x - 28 \\\\leq 6$, we accomplish this by adding 28 to both sides. This action moves the constant term from the left side to the right side, further simplifying the inequality and bringing us closer to solving for 'x'.

Adding 28 to both sides, we have:

$49x - 28 + 28 \\\\leq 6 + 28$

This simplifies to:

$49x \\\\leq 34$

By isolating the constant terms, we've created a situation where the variable term is on one side and the constant term is on the other. This is a critical step in solving for 'x' because it allows us to divide by the coefficient of 'x' and determine the solution set.

4. Solve for x

With the variable term and constant term isolated, we can now solve for 'x' in the inequality $49x \\\\leq 34$. To do this, we divide both sides of the inequality by the coefficient of 'x', which is 49. This operation will give us the range of values for 'x' that satisfy the inequality.

Dividing both sides by 49, we get:

$\\\\frac{49x}{49} \\\\leq \\\\frac{34}{49}$

This simplifies to:

$x \\\\leq \\\\frac{34}{49}$

Therefore, the solution to the inequality is $x \\\\leq \\\\frac{34}{49}$. This means that any value of 'x' that is less than or equal to $\\\\frac{34}{49}$ will satisfy the original inequality. We have now successfully solved the inequality, and the next step is to represent this solution graphically on a number line.

Graphing the Solution on a Number Line

Graphing the solution of an inequality on a number line provides a visual representation of the solution set. For the inequality $x \\\\leq \\\\frac{34}{49}$, we need to represent all values of 'x' that are less than or equal to $\\\\frac{34}{49}$. Here's how we do it:

1. Locate $\\\\frac{34}{49}$ on the Number Line:

First, we need to find the approximate location of $\\\\frac{34}{49}$ on the number line. Since $\\\\frac{34}{49}$ is less than 1 but greater than 0, it will be located between 0 and 1. You can estimate its position more precisely by converting it to a decimal (approximately 0.69) and marking that point on the number line.

2. Use a Closed Circle or Bracket:

Because the inequality includes "equal to" ($\\\\\leq$), we use a closed circle or a bracket at $\\\\frac{34}{49}$ to indicate that this value is included in the solution set. A closed circle is a filled-in circle, while a bracket is a square bracket facing the direction of the solution.

3. Shade the Line to the Left:

Since the solution includes all values of 'x' that are less than $\\\\frac{34}{49}$, we shade the line to the left of the closed circle or bracket. This shaded portion represents all the numbers that satisfy the inequality.

By following these steps, you can accurately represent the solution $x \\\\leq \\\\frac{34}{49}$ on a number line. This visual representation is a powerful tool for understanding the range of values that satisfy an inequality.

Common Mistakes to Avoid

When solving and graphing inequalities, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. Here are some key mistakes to watch out for:

1. Forgetting to Flip the Inequality Sign:

One of the most critical rules to remember is that when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if you have $-2x < 4$, dividing both sides by -2 requires you to change the sign to $x > -2$. Forgetting to do this will result in an incorrect solution.

2. Incorrectly Distributing:

When distributing a number across parentheses, ensure that you multiply it by every term inside the parentheses. For instance, in the expression $3(x - 2)$, you should distribute the 3 to both 'x' and '-2', resulting in $3x - 6$. A common mistake is to only multiply by the first term, leading to an incorrect simplification.

3. Errors in Arithmetic:

Simple arithmetic errors can derail the entire solution process. Double-check each step of your calculations, including addition, subtraction, multiplication, and division. Even a small mistake can lead to a wrong answer.

4. Misinterpreting the Inequality Sign:

It's essential to correctly interpret what each inequality sign means. Remember:

• < means "less than"

• means "greater than"

• $\\\\\leq$ means "less than or equal to"
• $\\\\\geq$ means "greater than or equal to"

Using the wrong sign can lead to graphing the solution set incorrectly.

5. Graphing Errors:

When graphing on a number line, remember to use a closed circle (or bracket) for $\\\\\leq$ and $\\\\\geq$ to indicate that the endpoint is included in the solution set. Use an open circle for < and > to show that the endpoint is not included. Also, make sure you shade the correct side of the number line based on the inequality sign.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving and graphing inequalities.

Real-World Applications

Inequalities aren't just abstract mathematical concepts; they have numerous real-world applications. Understanding how to solve and graph inequalities can help you make informed decisions in various situations. Here are a few examples:

1. Budgeting and Finance:

Inequalities are often used in budgeting to represent spending limits. For instance, if you have a budget of $50 for groceries, you can represent your spending with the inequality $x \\\\leq 50$, where 'x' is the amount you spend. This helps you ensure that your expenses stay within your budget.

2. Speed Limits and Traffic Laws:

Speed limits on roads are expressed as inequalities. A sign that says "Maximum Speed 65 mph" means that your speed (s) must satisfy the inequality $s \\\\leq 65$. Similarly, minimum speed limits can also be represented using inequalities.

3. Health and Fitness:

Inequalities can be used to define healthy ranges for various health metrics. For example, a healthy blood pressure reading might be represented as $120 \\\\leq systolic \\\\leq 140$. This helps individuals understand whether their health indicators are within the recommended range.

4. Manufacturing and Quality Control:

In manufacturing, inequalities are used to set tolerances for product dimensions. For instance, the diameter (d) of a manufactured bolt might need to satisfy the inequality $4.95 \\\\leq d \\\\leq 5.05$ mm to meet quality standards. This ensures that products are within acceptable limits.

5. Temperature Ranges:

Weather forecasts often use inequalities to describe temperature ranges. For example, a forecast might state that the temperature (T) will be between 20°C and 25°C, which can be written as $20 \\\\leq T \\\\leq 25$.

6. Resource Allocation:

Inequalities are used in resource allocation to ensure that resources are used efficiently and fairly. For instance, a company might have a limited budget for marketing and needs to allocate it among different channels while staying within the budget constraint.

These examples illustrate that inequalities are a powerful tool for representing and solving real-world problems involving constraints, limits, and ranges. By mastering the concepts of inequalities, you can better understand and navigate various aspects of daily life.

Conclusion

In conclusion, we have successfully solved the inequality $(35x - 14) \\\\leq \\\\frac{21x}{2} + 3$ and found the solution to be $x \\\\leq \\\\frac{34}{49}$. We also discussed how to graph this solution on a number line, highlighting the importance of using a closed circle or bracket and shading the correct side of the line. Additionally, we covered common mistakes to avoid and explored real-world applications of inequalities.

Understanding inequalities is a crucial skill in mathematics and has practical applications in various fields. By following the steps outlined in this guide and practicing regularly, you can confidently solve and graph inequalities, applying this knowledge to real-world scenarios.