Solve The Inequality: -12 + D <= -11

When we're faced with an inequality like -12 + d ≤ -11, our goal is to find the values of the variable, in this case 'd', that make the statement true. Think of an inequality as a balancing scale; we want to keep both sides balanced while isolating the variable. The key is to perform the same operation on both sides of the inequality to maintain the truth of the statement. So, to tackle -12 + d ≤ -11, we first want to get 'd' by itself. The first step is to eliminate the '-12' that's on the same side as 'd'. We can do this by adding 12 to both sides of the inequality. Remember, whatever you do to one side, you must do to the other to keep the inequality valid. So, we'll add 12 to the left side: -12 + d + 12. This simplifies to just 'd'. Now, we do the same to the right side: -11 + 12. This sum equals 1. So, our inequality now reads d ≤ 1. This means any value of 'd' that is less than or equal to 1 will satisfy the original inequality. This is the solution set for our inequality.

Now, let's consider the options provided to see which one fits our solution: d ≤ 1. We have four choices: A. d = -7, B. d = 11, C. d = 12, and D. d = 9. We need to check each of these values against our derived inequality, d ≤ 1. Let's start with option A, where d = -7. Is -7 less than or equal to 1? Yes, it is! So, d = -7 is a potential solution. Now, let's look at option B, where d = 11. Is 11 less than or equal to 1? No, 11 is much greater than 1. Therefore, d = 11 is not a solution. Moving on to option C, we have d = 12. Is 12 less than or equal to 1? Again, no. 12 is significantly larger than 1, so this is not a solution either. Finally, let's examine option D, where d = 9. Is 9 less than or equal to 1? Certainly not. 9 is greater than 1. So, d = 9 is also not a solution. Based on this analysis, only d = -7 satisfies the condition d ≤ 1. Therefore, option A is the correct answer to the question of which of the following is a solution to the inequality -12 + d ≤ -11.

Understanding Inequalities: A Deeper Dive

Let's expand on the concept of inequalities and why understanding them is crucial in mathematics and beyond. An inequality is a mathematical statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which assert that two expressions are equal, inequalities describe a range of possible values. This range is fundamental in many areas of study, from economics and engineering to computer science and statistics. For instance, in physics, you might deal with inequalities describing the range of possible velocities a particle can have. In finance, you might use inequalities to set profit margins or budget constraints. The ability to manipulate and solve inequalities is therefore a core mathematical skill.

Our specific inequality, -12 + d ≤ -11, is a linear inequality in one variable. The process of solving it involves algebraic manipulation aimed at isolating the variable 'd'. We achieved this by adding 12 to both sides. This operation preserves the inequality's truth because adding the same positive number to both sides does not change the relative order of the two expressions. If we had been dealing with multiplication or division by a negative number, we would have had to reverse the inequality sign. However, in this case, the operation was simple addition. The result, d ≤ 1, tells us that any number that is 1 or smaller will make the original statement true. We can visualize this on a number line. We would draw a solid circle at 1 (to include 1 itself) and shade the line to the left, indicating all numbers less than 1.

When we test the given options, we are essentially checking if specific points fall within this solution set. For d = -7, we substitute it back into the original inequality: -12 + (-7) ≤ -11. This becomes -19 ≤ -11, which is true. For d = 11, we get -12 + 11 ≤ -11, resulting in -1 ≤ -11, which is false. For d = 12, we have -12 + 12 ≤ -11, which simplifies to 0 ≤ -11, also false. Lastly, for d = 9, we get -12 + 9 ≤ -11, leading to -3 ≤ -11, which is incorrect. This step-by-step verification confirms that d = -7 is the only valid solution among the choices.

It's important to remember that inequalities can have infinitely many solutions, represented by a range. Our inequality d ≤ 1 represents an infinite set of numbers, including all negative integers, zero, and all numbers between zero and one, as well as fractions and decimals within that range. When presented with multiple-choice options, we are simply looking for one value that belongs to this infinite set. This skill is foundational for more complex mathematical concepts, such as graphing regions in coordinate planes defined by multiple inequalities (systems of inequalities) or understanding the domain and range of functions.

Practical Applications of Solving Inequalities

Beyond the classroom, the ability to solve and interpret inequalities is surprisingly practical. Consider a scenario where you are managing a budget. You might have an inequality representing your spending limit, such as Total Expenses ≤ Budgeted Amount. If your current expenses plus a new purchase exceed your budget, you know that purchase is not feasible. This is a direct application of inequality principles. In manufacturing, quality control often involves ensuring that a product's dimensions fall within a certain range, defined by an upper and lower bound – essentially, two inequalities working together. For example, the length of a component might need to be 10 cm ≤ Length ≤ 10.1 cm. If a manufactured part has a length outside this range, it's rejected.

In the context of our specific problem, -12 + d ≤ -11, the solution d ≤ 1 can be interpreted in various ways depending on what 'd' represents. If 'd' represents a change in temperature, it means the temperature change must be 1 degree Celsius or less to meet a certain condition. If 'd' represents the number of additional items you can produce within a certain timeframe, it means you can produce at most 1 additional item. The process of solving inequalities is not just about manipulating symbols; it's about understanding constraints and possibilities. The isolation of the variable 'd' provides a clear boundary – in this case, the number 1 – that defines what values are acceptable.

When we are given multiple-choice answers, as in this problem, the task becomes identifying which specific number falls into the solution set. This is often a preliminary step before tackling more complex problems where you might need to graph the solution set or use it in further calculations. The options given (d=-7, d=11, d=12, d=9) are distinct values. By substituting each one back into the original inequality, we rigorously check their validity. The inequality -12 + d ≤ -11 is a simple linear inequality, and its solution set is a half-infinite interval on the number line. Testing each option confirms that only d = -7 lies within this interval, making it the correct answer.

Furthermore, understanding the difference between 'less than' (<) and 'less than or equal to' (≤) is vital. If the inequality had been -12 + d < -11, the solution would be d < 1. In this case, d = 1 would not be a solution. The presence of the