Solving Inequalities: Find The True Values

Hey everyone! Today, we're diving into the world of inequalities. Specifically, we'll tackle a problem where we need to find the values that make a given inequality true. This is a fundamental concept in mathematics, and understanding it is key to solving more complex problems. Let's get started, shall we?

The Inequality Challenge

Our inequality is: $-\frac{1}{3}(57+21 x)-13<14 x+31$

Our mission, should we choose to accept it (and we do!), is to figure out which of the provided values make this statement a reality. The process involves simplifying the inequality and isolating the variable, which in this case is 'x'. This is similar to solving equations, but with a slight twist: instead of finding a single solution, we're looking for a range of values. The goal is to determine the values that satisfy the inequality. To do this, we'll need to use our algebra skills to manipulate the inequality and solve for x. Remember, our ultimate aim is to find the values of x that make the inequality true. Let's break this down step by step, shall we?

First, we'll tackle the left side of the inequality. We begin by distributing the $-\frac{1}{3}$ across the terms inside the parentheses. This means multiplying $-\frac{1}{3}$ by both 57 and 21x. So, $-\frac{1}{3}*57* results in -19, and *$-\frac{1}{3}21x simplifies to -7x. Thus, the left side of the inequality becomes -19 - 7x. Now, the inequality looks like this: -19 - 7x - 13 < 14x + 31. Next, combine like terms on the left side. -19 and -13 combine to become -32. Therefore, our inequality now reads: -32 - 7x < 14x + 31. The next step is to get all the 'x' terms on one side of the inequality and the constant terms on the other side. To do this, we can add 7x to both sides. This eliminates the -7x on the left side, giving us -32 < 21x + 31. Then, subtract 31 from both sides to isolate the term with x. This leaves us with -63 < 21x. Finally, we can solve for x by dividing both sides by 21. This gives us x > -3. So, the solution to the inequality is x > -3. This means that any value of x greater than -3 will make the original inequality true. We have to be meticulous while solving this. So that we don't miss any steps, or end up with a wrong result. Now, let us check which of the given values satisfy this condition.

Checking the Values

Now that we've solved the inequality, let's check which of the provided values fit the bill. Remember, we're looking for values of x that are greater than -3.

Here are the values we need to evaluate:

• $-\frac{13}{2}$* (which is -6.5)
• -3
• 0
• 3
• $\frac{13}{2}$* (which is 6.5)

Let's go through them one by one:

• $-\frac{13}{2}$ (-6.5):* This is less than -3, so it does not satisfy the inequality.
• -3: This is equal to -3, so it does not satisfy the inequality (we need values greater than -3).
• 0: This is greater than -3, so it does satisfy the inequality.
• 3: This is greater than -3, so it does satisfy the inequality.
• $\frac{13}{2}$ (6.5):* This is greater than -3, so it does satisfy the inequality.

So, the values that make the inequality true are 0, 3, and 6.5. Now let us try to solve another similar example. Also, it is very important to get the correct answer. We need to be able to apply the same concept when we come across similar problems. The more you solve, the better the concept is. Practice is very essential if you want to be perfect.

More Examples

Let's try a few more examples to cement our understanding of inequalities.

Example 1: Solve and find the values that satisfy the inequality: 2(x + 4) - 5 > 3x - 1.

First, distribute the 2 on the left side: 2x + 8 - 5 > 3x - 1. Then, simplify the left side: 2x + 3 > 3x - 1. Subtract 2x from both sides: 3 > x - 1. Finally, add 1 to both sides: 4 > x, or x < 4. Any value of x less than 4 satisfies the inequality.

Example 2: Solve: \frac{1}{2}(6x - 4) \leq 8.

Distribute the \frac1}{2} 3x - 2 \leq 8. Add 2 to both sides: 3x \leq 10. Divide both sides by 3: x \leq \frac{10{3}. So, any value of x less than or equal to \frac{10}{3} satisfies the inequality. Remember, when multiplying or dividing both sides of an inequality by a negative number, you must flip the inequality sign. Let's create an example for this type.

Example 3: Solve and find the values that satisfy the inequality: -3x + 6 < -9.

Subtract 6 from both sides: -3x < -15. Divide both sides by -3. Since we're dividing by a negative number, we flip the inequality sign: x > 5. Any value of x greater than 5 satisfies the inequality. You see how different the concept is. You need to remember the rule.

Conclusion

Alright, guys, we've covered the basics of solving inequalities and finding the values that make them true. Remember, the key is to isolate the variable and pay close attention to the direction of the inequality sign. Keep practicing, and you'll become a pro in no time! Always remember to double-check your work, and don't be afraid to ask for help if you get stuck. Happy solving!