Problem 52 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ [-2+(-7)]+[-11+22] $$

Step-by-Step Solution

Answer

The result of the expression is 2.

Step 1: Simplify the First Bracket

First, look at the expression inside the first bracket, which is \[-2 + (-7)\]. Combine the numbers by adding them: \(-2 + (-7) = -2 - 7 = -9\).

Step 2: Simplify the Second Bracket

Next, simplify the expression in the second bracket:\[-11 + 22\].Add the numbers: \(-11 + 22 = 11\).

Step 3: Add the Results from Both Brackets

Now add the results from the two brackets together:\(-9 + 11 = 2\).

Key Concepts

Adding Integers

Adding integers involves understanding how numbers, both positive and negative, interact with each other. To add integers effectively:

• If both integers are positive, simply add their values just like regular numbers. For example, adding 3 and 5 gives you 8.
• If both integers are negative, add their absolute values and then place a negative sign in front of the result. For example, adding -4 and -6 results in -4 - 6 = -10.
• If one integer is positive and the other is negative, find the difference between the absolute values, then sign the result with the sign of the larger absolute value. For instance, adding 9 and -3 results in 9 - 3 = 6. Here, since 9 has a larger absolute value than -3, the result is positive.

This approach helps one to master understanding and solving equations involving various combinations of positive and negative integers.

Negative Numbers

Negative numbers can sometimes be a bit tricky when working with expressions. Understanding how they behave in arithmetic operations is crucial:

• Negative numbers lie to the left of zero on the number line, meaning they are less than zero.
• When you add a negative number, you actually subtract its absolute value. For instance, adding -5 to a number is the same as subtracting 5 from that number.
• Conversely, subtracting a negative number is like adding its absolute value. Subtracting -3 from a number increases the value by 3.

These insights into negative numbers aid in correctly solving expressions that include them. Remembering these pointers can prevent common errors, especially when the expressions become more complex.

Simplifying Expressions

Simplifying expressions involves breaking them down into their simplest form. This makes calculations easier and more manageable:

• Start by simplifying expressions within brackets or parentheses first, as these calculations take priority.
• Combine like terms, which are terms in the expression that have the same variable and power. In cases without variables, like integers, simply sum them up as illustrated in the exercise.
• Be cautious with signs; always apply them correctly as you combine terms.

For example, in the given exercise, simplifying ":[-2+(-7)]+[-11+22]" requires handling the expressions step by step. Start with the brackets first as they carry the basic units of calculation. This approach ensures clarity and accuracy when simplifying any mathematical expressions.