Question

Evaluate the expression. $$ 14+|-11|-10 $$

Short answer

The expression evaluates to 15.

Step-by-step solution

Solve Absolute Value

The simplest task is to resolve the absolute value operation. The absolute value of a number is its distance from zero, so it's always positive. From this, we know that |-11| equals 11.

Perform Addition and Subtraction operations

Once the absolute value has been solves, we can proceed with the remainder of the operations. Using the values on either side of it, complete the addition and subtraction operations: 14 + 11 - 10.

Calculate final result

Now, just add 14 to 11 to get 25, then subtract 10 from 25 to get 15.

Key concepts

Absolute Value

Understanding the concept of absolute value is essential in mathematics as it denotes the distance of a number from zero on a number line, regardless of the direction. It is always a non-negative value. For example, both 5 and -5 have an absolute value of 5, symbolized in mathematics as |5| and |-5| respectively. In the given exercise:

$$ 14 + |-11| - 10 $$

the absolute value of -11 is treated first. Think of it as a reassurance that no matter if you've misplaced your number on the positive or negative side of the number line, the absolute value will safely guide it back to the positive side. So, |-11| simply becomes 11.

Order of Operations

When evaluating expressions involving multiple operations such as addition, subtraction, and absolute values, the order of operations provides a standardized method to ensure consistent results. This is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

The absolute value can be thought of as a 'parentheses-like' operation in that it should be resolved first before addressing other operations. Afterward, move from left to right, combining like terms or performing other addition and subtraction as the exercise requires. The clear structure of PEMDAS helps prevent confusion and ensures accuracy.

Addition and Subtraction in Algebra

Upon clarifying the absolute values and following the order of operations, addition and subtraction within algebra are straightforward. These operations are about combining or taking away quantities. In algebra, it is just a matter of combining like terms and simplifying expressions down to their simplest form. For the given exercise:

$$ 14 + 11 - 10 $$

Once the absolute value is calculated, the expression simplifies to standard addition and subtraction, which should be executed from left to right. This process is slightly akin to balancing your checkbook; you're adding deposits (14 + 11) and subtracting withdrawals (-10) to calculate your final balance. When simplified, the expression yields 15, which is the simplest form, akin to knowing your exact account balance after all transactions are accounted for.