Master Compound Inequalities: Solve And Graph With Ease

When we talk about solving inequalities, we're essentially figuring out the range of numbers that make a certain mathematical statement true. It's like finding all the possible scores a student could get on a test to pass, given certain conditions. Now, sometimes, these conditions come in pairs, and that's where compound inequalities enter the picture. These are two or more inequalities joined together, usually by the words "and" or "or." Today, we're going to dive deep into solving a specific compound inequality: $x+7>3$ and $x-5 \leq-1$. We'll not only find the solution for $x$ but also understand how to represent this solution visually on a graph. It's a fundamental skill in algebra that opens doors to understanding more complex mathematical concepts and real-world applications where ranges and conditions are key.

Let's break down the inequality $x+7>3$ and $x-5 \leq-1$ step by step. This compound inequality consists of two separate inequalities that both must be true simultaneously because they are connected by the word "and." Think of it as needing to meet two requirements to achieve a goal; both must be satisfied. First, let's tackle the inequality $x+7>3$. To isolate $x$, we need to perform the opposite operation of adding 7, which is subtracting 7, from both sides of the inequality. So, we have $x+7-7 > 3-7$, which simplifies to $x > -4$. This tells us that any value of $x$ that is greater than -4 will satisfy this first condition. We represent this on a number line with an open circle at -4 (because $x$ is not equal to -4) and an arrow pointing to the right, indicating all numbers larger than -4. This is a crucial first piece of our puzzle.

Now, let's move on to the second inequality in our compound statement: $x-5 \leq-1$. Similar to the first inequality, our goal here is to get $x$ by itself. The operation currently being applied to $x$ is subtraction of 5. To reverse this, we'll add 5 to both sides of the inequality. This gives us $x-5+5 \leq-1+5$, which simplifies to $x \leq 4$. This second part of our compound inequality tells us that $x$ must be less than or equal to 4. On a number line, this is represented by a closed circle at 4 (because $x$ can be equal to 4) and an arrow pointing to the left, indicating all numbers smaller than or equal to 4.

So, we have two conditions for $x$: $x > -4$ and $x \leq 4$. Since the original compound inequality uses the word "and," we are looking for the values of $x$ that satisfy both conditions at the same time. This means we need to find the overlap between the solution sets of the two individual inequalities. The first inequality, $x > -4$, includes all numbers to the right of -4 on the number line. The second inequality, $x \leq 4$, includes all numbers to the left of and including 4 on the number line. The "and" condition requires us to find where these two ranges intersect. Graphically, this means we are looking for the segment of the number line that is both to the right of -4 and to the left of or at 4. This intersection forms a continuous interval.

To visualize this, imagine drawing both solutions on the same number line. You'd have an open circle at -4 with an arrow going right, and a closed circle at 4 with an arrow going left. The region where both arrows (or the shaded areas they represent) overlap is our solution. This overlap starts just after -4 (since $x > -4$) and ends at 4 (since $x \leq 4$). Therefore, the compound inequality $x+7>3$ and $x-5 \leq-1$ is satisfied by all values of $x$ that are greater than -4 AND less than or equal to 4. We can write this combined solution as $-4 < x \leq 4$. This notation elegantly captures both conditions in a single, concise expression. It tells us that $x$ can be any number strictly between -4 and 4, including 4 itself.

Let's consider the given options to identify the correct solution and its graphical representation. We found that the solution to $x+7>3$ is $x > -4$, and the solution to $x-5 \leq-1$ is $x \leq 4$. Because the inequalities are connected by "and," we need the values of $x$ that satisfy both. This leads to the compound inequality $-4 < x \leq 4$. Now, let's examine the choices provided:

• A. Solution: $x \geq-4$ and $x<4$: This is incorrect because our first inequality resulted in $x > -4$, not $x \geq-4$. Also, the second inequality is $x \leq 4$, not $x<4$.
• B. Solution: $x<-4$ or $x \geq 4$: This option uses "or," which means we'd be looking for numbers that satisfy either inequality, not necessarily both. Our original problem uses "and." Furthermore, the ranges themselves are not correct based on our calculations.
• C. Solution: $x>-4$ and $x \leq 4$: This matches our derived solution perfectly. The graph of this solution would show an open circle at -4 (since $x$ is strictly greater than -4) and a closed circle at 4 (since $x$ is less than or equal to 4), with the region between them shaded. This represents all numbers that are greater than -4 and less than or equal to 4.
• D. Solution: $x>-4$ and $x \leq 4$: This option is identical to option C and also correctly represents our solution. In a multiple-choice scenario, if two options are identical and correct, it usually indicates a slight oversight in the question design, but both represent the right answer mathematically.

Therefore, the correct solution for the compound inequality $x+7>3$ and $x-5 \leq-1$ is indeed $x>-4$ and $x \leq 4$. The graphical representation would be a number line segment starting just after -4 (indicated by an open circle) and ending at 4 (indicated by a closed circle), with all numbers in between shaded. This visual depicts the intersection of the two solution sets, highlighting the values that satisfy both original conditions simultaneously. Understanding this process is key to mastering inequalities and their graphical interpretations, which are vital tools in various fields, including science, engineering, and economics.

If you're looking to further solidify your understanding of inequalities and their graphical solutions, exploring resources that delve into interval notation and set theory can be incredibly beneficial. These concepts provide a more formal framework for expressing and manipulating solution sets. For a deeper dive into the principles of algebra and problem-solving techniques, I highly recommend visiting websites like Khan Academy or consulting textbooks on pre-calculus and algebra.