Solving $-10 < -4 + 10m \leq -8$: A Step-by-Step Guide

Oct 10, 2025 by ADMIN 56 views

$Solving Inequalities: A Comprehensive Guide to $-10<-4+10m \leq -8$$

Hey guys! Today, we're diving into the world of inequalities. Inequalities might seem a bit intimidating at first, but trust me, once you grasp the basic concepts, they become quite manageable. We'll tackle the inequality $-10 < -4 + 10m \leq -8$ step by step. We will express our solution using interval notation and visualize it on a graph. So, grab your pencils and let's get started!

Understanding Inequalities

Before we jump into solving our specific inequality, let's quickly review what inequalities are all about. Unlike equations that have a single solution, inequalities represent a range of values. Common inequality symbols include:

< : Less than
> : Greater than
$\leq$ : Less than or equal to
$\geq$ : Greater than or equal to

When we solve an inequality, our goal is to isolate the variable (in this case, 'm') to determine the range of values that satisfy the inequality. Remember, whatever operation you perform on one part of the inequality, you must perform on all parts to maintain the balance.

Step-by-Step Solution of $-10<-4+10m \leq -8$

Now, let's break down the process of solving the inequality $-10 < -4 + 10m \leq -8$ . This type of inequality is called a compound inequality because it combines two inequalities into one statement. Our goal is to isolate 'm' in the middle section.

Step 1: Isolate the Term with 'm'

The first thing we want to do is get rid of the '-4' in the middle. To do this, we'll add 4 to all three parts of the inequality:

$-10 + 4 < -4 + 10m + 4 \leq -8 + 4$

This simplifies to:

$-6 < 10m \leq -4$

Step 2: Isolate 'm'

Now, we need to get 'm' all by itself. Since 'm' is being multiplied by 10, we'll divide all three parts of the inequality by 10:

$\frac{-6}{10} < \frac{10m}{10} \leq \frac{-4}{10}$

This simplifies to:

$-0.6 < m \leq -0.4$

Or, in fraction form:

$-\frac{3}{5} < m \leq -\frac{2}{5}$

Expressing the Solution in Interval Notation

Interval notation is a concise way to represent the range of values that satisfy the inequality. Here's how it works:

Parentheses '(' and ')' indicate that the endpoint is not included in the solution.
Brackets '[' and ']' indicate that the endpoint is included in the solution.
The symbol ' $\infty$ ' represents infinity.

For our inequality $-0.6 < m \leq -0.4$ , the solution in interval notation is:

$(-0.6, -0.4]$

This means that 'm' can be any value greater than -0.6 but less than or equal to -0.4. The parenthesis next to -0.6 indicates that -0.6 is not included, while the bracket next to -0.4 indicates that -0.4 is included.

Graphing the Solution

Visualizing the solution on a number line can be really helpful. Here's how to graph our solution:

Draw a number line: Draw a straight line and mark the numbers -0.6 and -0.4 on it.
Use parentheses or brackets: At -0.6, draw an open parenthesis '(' to indicate that -0.6 is not included in the solution. At -0.4, draw a closed bracket ']' to indicate that -0.4 is included.
Shade the region: Shade the region between -0.6 and -0.4 to represent all the values of 'm' that satisfy the inequality.

So, the graph will have an open parenthesis at -0.6, a closed bracket at -0.4, and the area between them shaded.

Key Considerations and Potential Pitfalls

While solving inequalities is pretty straightforward, there are a few things you need to watch out for:

Dividing or Multiplying by a Negative Number: If you divide or multiply an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you have -2m < 6, dividing by -2 gives you m > -3 (notice the flipped sign!).
Understanding Interval Notation: Make sure you understand the difference between parentheses and brackets. Using the wrong one can completely change the meaning of your solution.
Checking Your Solution: Always check your solution by plugging in a value from your solution set back into the original inequality. If the inequality holds true, you're on the right track!

Common Mistakes to Avoid

Forgetting to Flip the Inequality Sign: As mentioned earlier, this is a very common mistake when dividing or multiplying by a negative number.

Incorrectly Using Interval Notation: Mixing up parentheses and brackets is another frequent error. Double-check which endpoints are included in the solution.

Not Checking the Solution: Failing to verify your answer can lead to incorrect results. Always take a moment to plug a value back into the original inequality.

Real-World Applications

Inequalities aren't just abstract mathematical concepts; they have plenty of real-world applications. Here are a couple of examples:

Budgeting: Imagine you have a budget of $100 for groceries. If 'x' represents the amount you spend, you can write the inequality x $\leq$ 100. This helps you stay within your budget.
Temperature Ranges: Suppose a certain chemical reaction requires a temperature between 20°C and 30°C. If 'T' represents the temperature, you can write the inequality 20 < T < 30. This ensures the reaction occurs correctly.

Practice Problems

Want to test your understanding? Try solving these inequalities:

$3x + 5 > 14$
$-2 \leq 5 - x < 3$
$4(y - 1) \geq 8$

Conclusion

So, there you have it! Solving the inequality $-10 < -4 + 10m \leq -8$ involves isolating the variable 'm', expressing the solution in interval notation, and visualizing it on a graph. Remember, pay close attention to the rules for dividing by negative numbers and using interval notation correctly. Understanding inequalities is super useful, it will help you in math class and you will use it to solve problems in real life. Keep practicing, and you'll become a pro at solving inequalities in no time! Keep up the great work, guys! You've got this! Solving inequalities is a fundamental concept in algebra, and mastering it will undoubtedly benefit you in your future mathematical endeavors.