Question

Find the difference. $$ 7-|-1| $$

Short answer

So, the solution to the exercise \(7 - |-1|\) is 6.

Step-by-step solution

Understand the absolute value

The first step to solving this exercise is understanding how does the absolute value work. The absolute value of any number is the positive value of that number. Meaning that \(|-1|\) equals 1 due to its distance from zero is 1, regardless of whether it was negative or positive originally.

Substitute absolute value into the expression

Now substitute the absolute value of -1 (which was calculated to be 1 in the previous step) back into the expression. So, the expression \(7 + |-1|\) now becomes \(7 - 1\).

Perform the subtraction

Finally, perform the subtraction operation. The solution is \(7 - 1 = 6\).

Key concepts

Algebraic Expressions

Understanding algebraic expressions is a fundamental skill required to solve a wide range of mathematical problems. An algebraic expression is a collection of numbers, variables, and operations (such as addition and subtraction) that represent a specific value or set of values. In the given exercise, the expression is fairly simple: it consists of positive integers and the absolute value operation.

When dealing with algebraic expressions that contain absolute values, it's key to first evaluate inside the absolute signs before proceeding. This is because the absolute value sign acts like an operation that modifies the number or expression it encloses, converting it into its non-negative equivalent. Therefore, students need to isolate the absolute value from the rest of the expression, solve for it, and then carry on with the required operations - in this case, subtraction.

Here are some tips for simplifying algebraic expressions containing absolute values:

• Isolate the absolute value from other terms.
• Determine the absolute value independently.
• Substitute this non-negative value back into the expression.
• Carry out the remaining operations in the usual order.

Subtracting Integers

Subtracting integers might seem straightforward, but it can sometimes be trickier than it appears, especially when absolute values are involved. To subtract integers:

• First, observe the signs of the numbers involved. If they are both positive or both negative, subtract the smaller number from the larger one, and give the result the sign of the larger number.
• If the integers have different signs, add their absolute values together.
• Keep track of the signs; if you start with a positive number and subtract a negative one, you’re essentially adding a positive.

What is happening in our example is a common scenario in which a positive integer is being subtracted by the absolute value of a negative integer. Since the absolute value operation turns the negative integer into a positive one, you are simply subtracting a smaller positive integer from a larger one which yields a positive result. Here are tips to follow:

• Identify the absolute value and understand it as distance from zero.
• Turn the subtraction problem into an addition one if dealing with a 'minus a negative' scenario.
• Always work out absolute values first, then apply the subtraction.

Problem-Solving Steps

Breaking Down the Steps

Problem-solving in algebra can typically be tackled by a systematic approach, particularly helpful for exercises that require multiple operations. For the given problem, the steps break down as follows:

Step 1: Understand the Operation

Identify and understand each element of the exercise. Know what the absolute value signifies and remember that it will always result in a non-negative number. This understanding is crucial as it will affect how you manipulate the rest of the algebraic expression.

Step 2: Simplify the Expression

Before carrying out any operations, deal with the absolute value and any other aspects that can be simplified independently from the rest of the expression. In this case, you find the absolute value of -1, which is 1.

Step 3: Carry Out Operations

Once the absolute value is evaluated, integrate this value back into the expression and follow through with the remaining arithmetic. Here, you substitute |-1| with 1 to get 7 - 1 and then perform the subtraction.

Following these steps helps maintain order and prevents errors. It also ensures a consistent method when faced with more complex problems. Having a problem-solving strategy is key to success in algebra, as it provides a roadmap, reducing the anxiety that can come with facing an unfamiliar problem.