Solving Systems Of Inequalities: A Comprehensive Guide
Hey guys! Let's dive into the world of solving systems of inequalities. This topic is super important in algebra, and understanding it will help you ace your math tests. We will tackle two specific examples, breaking down each step to make sure you get a handle on the process. So, grab your pencils and let's get started. We are going to start with inequality systems and work our way through each part, explaining the ins and outs as we go along. Solving systems of inequalities involves finding the values that satisfy all inequalities in the system simultaneously. This often means finding a range of values, or in some cases, no solutions at all. The key is to simplify each inequality, isolate the variable, and then find the intersection of the solution sets. We'll be using this approach throughout our examples, ensuring that you understand the fundamental concepts. The first part will cover solving algebraic expressions. The second part will work through the steps involved in working out the solutions. Let's make sure we have a solid understanding of the basics. Let's get straight into it, shall we?
Example 1: Solving the First System of Inequalities
Our first problem involves the following system of inequalities:
a) { 3x-(x+1)² ≤ 5-(1-x)², (x-3)(x+3)>(x+7)(x-7). }
Let’s break it down step by step to solve these inequalities and find the values of x that satisfy both.
Step 1: Simplifying the First Inequality
The first inequality is: 3x - (x+1)² ≤ 5 - (1-x)². We need to expand the squared terms and simplify. This is where we need to remember our algebraic rules, like the FOIL method. Expanding (x+1)² gives us x² + 2x + 1. Similarly, expanding (1-x)² gives us 1 - 2x + x². Now, let's substitute these back into the inequality:
3x - (x² + 2x + 1) ≤ 5 - (1 - 2x + x²)
Simplifying further:
3x - x² - 2x - 1 ≤ 5 - 1 + 2x - x²
Combining like terms:
x - x² - 1 ≤ 4 + 2x - x²
Now, let's bring all the terms to one side. Notice that the x² terms cancel out:
x - 2x ≤ 4 + 1
-x ≤ 5
Finally, divide by -1 (and remember to flip the inequality sign when you divide by a negative number):
x ≥ -5
So, the solution to the first inequality is x ≥ -5.
Step 2: Simplifying the Second Inequality
Now, let's tackle the second inequality: (x-3)(x+3) > (x+7)(x-7). This is a great opportunity to practice your multiplication skills, or just use the difference of squares rule! Let's expand both sides. We have:
x² - 9 > x² - 49
Subtract x² from both sides:
-9 > -49
Wait a minute... This statement is not true. However, because the x² terms canceled out, there are no restrictions on x based on this inequality. This means that any value of x works for this second part of the system.
Step 3: Finding the Intersection of the Solutions
We found that the first inequality's solution is x ≥ -5, and the second inequality has no restrictions on x. This means that the solution to the system is the intersection of these two solution sets. Since the second inequality is true for all x, the overall solution will be determined by the first inequality. Therefore, the solution to the system of inequalities is x ≥ -5. The solution can be represented on a number line, and is a range of values, including -5 and extending to positive infinity. This is the final step. Congrats, you made it through the first example!
Example 2: Solving the Second System of Inequalities
Now, let’s solve the second system of inequalities. We can do this! Here's the second problem:
b) { (2y+15)/9 - (1-y)/5 ≥ y/3, 2y > (19-2y)/2 - (11-2y)/4. }
Let's break this down. We will work through each inequality separately and then find the intersection of the solution sets, just like we did before. Remember, the goal is to isolate the variable, which in this case is y.
Step 1: Simplifying the First Inequality
The first inequality is: (2y+15)/9 - (1-y)/5 ≥ y/3. To simplify this, we'll first eliminate the fractions. The least common multiple (LCM) of 9, 5, and 3 is 45. Multiply each term by 45:
45 * [(2y+15)/9] - 45 * [(1-y)/5] ≥ 45 * (y/3)
This simplifies to:
5(2y+15) - 9(1-y) ≥ 15y
Expand the terms:
10y + 75 - 9 + 9y ≥ 15y
Combine like terms:
19y + 66 ≥ 15y
Subtract 15y from both sides:
4y + 66 ≥ 0
Subtract 66 from both sides:
4y ≥ -66
Divide by 4:
y ≥ -66/4
Simplify the fraction:
y ≥ -33/2
So, the solution to the first inequality is y ≥ -33/2, which is equivalent to y ≥ -16.5.
Step 2: Simplifying the Second Inequality
Let's move on to the second inequality: 2y > (19-2y)/2 - (11-2y)/4. Again, let's get rid of the fractions by multiplying through by the least common multiple of the denominators, which is 4:
4 * 2y > 4 * [(19-2y)/2] - 4 * [(11-2y)/4]
This simplifies to:
8y > 2(19-2y) - (11-2y)
Expand the terms:
8y > 38 - 4y - 11 + 2y
Combine like terms:
8y > 27 - 2y
Add 2y to both sides:
10y > 27
Divide by 10:
y > 27/10
So, the solution to the second inequality is y > 27/10, which is equivalent to y > 2.7.
Step 3: Finding the Intersection of the Solutions
We found that y ≥ -16.5 and y > 2.7. The solution to the system is the intersection of these two solution sets. Since y must be greater than 2.7 to satisfy both inequalities, the overall solution to the system is y > 2.7. This can be represented on a number line, showing that the solution set includes all values greater than 2.7. We did it! We have solved the second system of inequalities.
Conclusion
Solving systems of inequalities can seem daunting at first, but with practice, it becomes much easier. Remember to simplify each inequality, isolate the variable, and then find the intersection of the solution sets. Always pay close attention to the inequality signs – especially when multiplying or dividing by a negative number. Keep practicing, and you'll become a pro at solving these types of problems. You've now seen two examples, and you should be able to tackle more complex examples. So go forth and solve those inequalities, guys!