Solving Absolute Value Inequality: |4x - 5| ≤ 7

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Let's dive into how to solve the absolute value inequality |4x - 5| ≤ 7. Absolute value inequalities might seem a bit tricky at first, but don't worry, guys! We'll break it down step by step so you can tackle these problems with confidence. Understanding how to handle absolute values is super useful in algebra and beyond, so let's get started!

Understanding Absolute Value Inequalities

Absolute value inequalities essentially ask you to find all values of a variable that make the distance between an expression and zero less than or equal to a certain number. In simpler terms, |x| ≤ a means that x must be between -a and a, inclusive. This is because absolute value measures distance from zero, so any number within 'a' units of zero satisfies the inequality. When you see an absolute value inequality, you're really dealing with two separate inequalities combined into one.

For example, if we have |x| ≤ 3, this means -3 ≤ x ≤ 3. Any number between -3 and 3 (including -3 and 3 themselves) will have an absolute value less than or equal to 3. Think about it: |-2| = 2, which is less than 3; |0| = 0, which is also less than 3; and |3| = 3, which is equal to 3. This concept is crucial for solving more complex absolute value inequalities.

The absolute value of a number is its distance from zero on the number line. It's always non-negative. Therefore, |x| is always greater than or equal to 0. When we solve absolute value inequalities, we are looking for the set of numbers that satisfy a certain condition related to their distance from zero. This usually involves splitting the problem into two cases to account for both positive and negative possibilities inside the absolute value.

In our specific problem, |4x - 5| ≤ 7, the expression inside the absolute value, 4x - 5, represents some distance from zero. We need to find all values of x that make this distance less than or equal to 7. This means that 4x - 5 must be between -7 and 7, inclusive. Setting up these two cases is the first major step in solving the inequality. Remember, guys, it's all about understanding the distance from zero!

Breaking Down the Inequality

To solve |4x - 5| ≤ 7, we need to break it down into two separate inequalities. This is because the expression inside the absolute value, 4x - 5, can be either positive or negative, and we need to account for both scenarios. So, we create two cases:

  1. Case 1: The expression inside the absolute value is positive or zero. In this case, 4x - 5 is already non-negative, so the absolute value doesn't change anything. We simply have: 4x - 5 ≤ 7

  2. Case 2: The expression inside the absolute value is negative. In this case, the absolute value changes the sign of the expression, so we need to consider the opposite of 4x - 5. This gives us: -(4x - 5) ≤ 7

Now, we have two separate inequalities that we can solve individually. Solving each of these will give us the range of x values that satisfy the original absolute value inequality. Remember, guys, it's like we're saying, "Okay, let's see what happens if this thing inside is positive, and then let's see what happens if it's negative." Breaking it down like this makes the problem much more manageable.

By considering both cases, we ensure that we capture all possible solutions to the inequality. Absolute value is all about distance, and distance is always positive or zero, so we need to think about both positive and negative versions of the expression inside the absolute value bars. It might seem a bit confusing at first, but with practice, it becomes second nature. So, let's move on and solve each of these inequalities!

Solving Case 1: 4x - 5 ≤ 7

Okay, guys, let's tackle the first case: 4x - 5 ≤ 7. This is a straightforward linear inequality, and we can solve it using basic algebraic steps. Our goal is to isolate x on one side of the inequality.

Step 1: Add 5 to both sides of the inequality.

This will help us get rid of the -5 on the left side. So we have:

4x - 5 + 5 ≤ 7 + 5

Simplifying, we get:

4x ≤ 12

Step 2: Divide both sides by 4.

This will isolate x and give us the solution for this case. Dividing both sides by 4, we get:

4x / 4 ≤ 12 / 4

Simplifying, we find:

x ≤ 3

So, for the first case, we have x ≤ 3. This means that any value of x that is less than or equal to 3 will satisfy the first part of our absolute value inequality. Remember, guys, this is only half the battle. We still need to solve the second case to get the complete solution.

It's important to note that we performed the same operations on both sides of the inequality to maintain its balance. Adding the same number to both sides or dividing both sides by the same positive number doesn't change the direction of the inequality. This is a fundamental principle in solving inequalities. So, now that we've solved the first case, let's move on to the second case and see what we get!

Solving Case 2: -(4x - 5) ≤ 7

Alright, let's jump into the second case: -(4x - 5) ≤ 7. This case deals with the scenario where the expression inside the absolute value is negative. Remember, guys, we need to handle this negative sign carefully.

Step 1: Distribute the negative sign.

First, we distribute the negative sign to both terms inside the parentheses:

-4x + 5 ≤ 7

Step 2: Subtract 5 from both sides.

Now, let's subtract 5 from both sides to isolate the term with x:

-4x + 5 - 5 ≤ 7 - 5

Simplifying, we get:

-4x ≤ 2

Step 3: Divide both sides by -4. Remember to flip the inequality sign!

Since we're dividing by a negative number, we need to flip the inequality sign. This is a crucial step! Dividing both sides by -4, we get:

-4x / -4 ≥ 2 / -4

Simplifying, we find:

x ≥ -1/2

So, for the second case, we have x ≥ -1/2. This means that any value of x that is greater than or equal to -1/2 will satisfy the second part of our absolute value inequality. Remember, guys, flipping the inequality sign when dividing by a negative number is super important! If you forget to do that, you'll end up with the wrong solution.

Now that we've solved both cases, we need to combine the results to find the complete solution to the original absolute value inequality. Let's see how to do that!

Combining the Solutions

Okay, guys, we've solved both cases of our absolute value inequality. Now it's time to put the pieces together and find the overall solution. We found that:

  • Case 1: x ≤ 3
  • Case 2: x ≥ -1/2

To satisfy the original inequality |4x - 5| ≤ 7, x must satisfy both of these conditions. In other words, x must be less than or equal to 3 AND greater than or equal to -1/2. We can write this as a compound inequality:

-1/2 ≤ x ≤ 3

This means that x can be any value between -1/2 and 3, including -1/2 and 3 themselves. This is our final solution!

We can also represent this solution graphically on a number line. Draw a number line, mark -1/2 and 3, and shade the region between them. Use closed circles (or brackets) at -1/2 and 3 to indicate that these values are included in the solution. This visual representation can be helpful for understanding the range of values that satisfy the inequality.

So, to recap, solving absolute value inequalities involves breaking them down into two cases, solving each case separately, and then combining the solutions. Remember to flip the inequality sign when dividing by a negative number, and be careful with the negative signs! With practice, you'll become a pro at solving these types of problems.

Final Answer

Therefore, the solution to the absolute value inequality |4x - 5| ≤ 7 is:

-1/2 ≤ x ≤ 3

This means that any value of x between -1/2 and 3, inclusive, will satisfy the inequality. Great job, guys! You've successfully solved an absolute value inequality. Keep practicing, and you'll master these types of problems in no time! Remember, the key is to understand the concept of absolute value and to break down the problem into manageable steps. Now go out there and conquer more math challenges!