Solving Inequalities Translating Statements Into Inequalities
Hey guys! Today, we're diving into the exciting world of inequalities and how to translate word problems into mathematical expressions. This is a crucial skill in algebra, and once you get the hang of it, you'll be solving these problems like a pro. We'll break down each statement, turn it into an inequality, and then solve it step by step. So, let's get started!
Understanding Inequalities
Before we jump into the problems, let's quickly recap what inequalities are. Unlike equations that show equality (using the "=" sign), inequalities show relationships where values are not necessarily equal. We use symbols like:
- < (less than)
- > (greater than)
- ≤ (less than or equal to)
- ≥ (greater than or equal to)
Understanding these symbols is key to translating word problems accurately. When we see phrases like "at most," "no more than," or "is less than or equal to," we know we're dealing with a "≤" inequality. Similarly, phrases like "at least," "no less than," or "is greater than or equal to" indicate a "≥" inequality. Keeping these in mind will help us convert the word sentences into mathematical inequalities.
Problem 1 The quotient of a number and 4 is at most 5.
Let's start with the first statement "The quotient of a number and 4 is at most 5." The key here is to break it down piece by piece. First, we need to identify the unknown number, and let's call it x. The term "quotient" tells us we're dealing with division. So, "the quotient of a number and 4" can be written as x / 4. Now, the phrase "is at most 5" means that the result can be 5 or less than 5. This translates to the inequality symbol "≤". So, putting it all together, the inequality is:
x / 4 ≤ 5
Now, let's solve this inequality. To isolate x, we need to undo the division by 4. We do this by multiplying both sides of the inequality by 4. Remember, whatever you do to one side of an inequality, you must do to the other to maintain the balance. This gives us:
(x / 4) * 4 ≤ 5 * 4
Simplifying, we get:
x ≤ 20
So, the solution to the inequality is x ≤ 20. This means any number that is 20 or less will satisfy the original statement. To double-check, we can pick a number less than or equal to 20, like 16. Plugging this back into our original inequality, we get 16 / 4 ≤ 5, which simplifies to 4 ≤ 5. This is true, so our solution checks out. What about trying 20 itself? 20 / 4 ≤ 5 becomes 5 ≤ 5, which is also true because 5 is equal to 5. Remember, the "less than or equal to" symbol includes the possibility of equality.
Problem 2 A number divided by 7 is less than -3.
Next up, we have the statement "A number divided by 7 is less than -3." Again, let's break it down. Let's use y to represent "a number" this time, just to mix things up. "Divided by 7" means we have y / 7. The phrase "is less than -3" is straightforward – it translates to the "<" symbol followed by -3. So, the inequality is:
y / 7 < -3
To solve for y, we need to get rid of the division by 7. Just like before, we multiply both sides of the inequality by 7:
(y / 7) * 7 < -3 * 7
This simplifies to:
y < -21
So, the solution is y < -21. This means any number less than -21 will make the original statement true. Let's test this with a number less than -21, say -28. Plugging this back into the original inequality, we have -28 / 7 < -3, which simplifies to -4 < -3. This is true since -4 is indeed less than -3. Remember, when dealing with negative numbers, the further away from zero you go in the negative direction, the smaller the number is.
Problem 3 Six times a number is at least -24.
Now, let's tackle the statement "Six times a number is at least -24." Let's use z to represent "a number" this time. "Six times a number" translates to 6 * z, or simply 6z. The phrase "is at least -24" means the result can be -24 or greater than -24. This corresponds to the "≥" symbol. So, our inequality is:
6z ≥ -24
To solve for z, we need to isolate it by dividing both sides of the inequality by 6:
(6z) / 6 ≥ -24 / 6
Simplifying, we get:
z ≥ -4
So, the solution is z ≥ -4. This means any number that is -4 or greater will satisfy the original statement. Let's test this with a number greater than -4, such as 0. Plugging it in, we get 6 * 0 ≥ -24, which simplifies to 0 ≥ -24. This is true because 0 is greater than -24. Now, let's try -4 itself. 6 * (-4) ≥ -24 simplifies to -24 ≥ -24, which is also true because -24 is equal to -24. This confirms that our solution is correct.
Problem 4 The product of -2 and a number is greater than 30.
Finally, let's look at the statement "The product of -2 and a number is greater than 30." Let's use w to represent "a number." The phrase "the product of -2 and a number" means -2 multiplied by w, which we write as -2w. "Is greater than 30" simply translates to the ">" symbol followed by 30. So, the inequality is:
-2w > 30
Here's where things get a little trickier. To solve for w, we need to divide both sides of the inequality by -2. But there's a crucial rule to remember when dealing with inequalities: when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers on the number line.
So, when we divide both sides by -2, we get:
(-2w) / -2 < 30 / -2
Notice that the "greater than" sign (>) has flipped to a "less than" sign (<). Simplifying, we have:
w < -15
So, the solution is w < -15. This means any number less than -15 will satisfy the original statement. Let's test this with -16. Plugging it back into the original inequality, we get -2 * (-16) > 30, which simplifies to 32 > 30. This is true. Remember, the rule about flipping the inequality sign is super important, so don't forget it!
Key Takeaways
Alright, guys, let's recap what we've learned today. Translating word problems into inequalities involves breaking down the statements into smaller, manageable parts. We identify the unknown, represent it with a variable, and then translate the phrases into mathematical symbols and operations. Key phrases like "at most," "at least," "less than," and "greater than" help us determine the correct inequality symbol to use.
Solving inequalities is similar to solving equations, but remember the golden rule: if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is crucial for getting the correct solution.
And always, always, always check your solution by plugging a value from your solution set back into the original inequality. This will help you catch any mistakes and ensure you've got the right answer.
Practice Makes Perfect
The best way to get comfortable with translating and solving inequalities is to practice. Work through lots of different examples, and don't be afraid to make mistakes – that's how we learn! The more you practice, the easier it will become to recognize the key phrases and translate them into mathematical expressions.
So, go out there and conquer those inequalities! You've got this!
