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Chapter 15: Problem 86

Remove parentheses and simplify. $$ 2[2 a+1-(3 a-6)] $$

Short Answer

Expert verified
-2 a + 14

Step by step solution

01

Distribute inside the parentheses

Simplify the expression inside the parentheses first: 2 a + 1 - (3 a - 6).
02

Simplify inside the parentheses

Distribute the negative sign inside the parentheses: 2 a + 1 - 3 a + 6.
03

Combine like terms

Combine like terms: (2 a - 3 a) + (1 + 6) = - a + 7.
04

Distribute the outer coefficient

Now distribute the 2 to both terms inside the brackets: 2(- a + 7) = -2 a + 14.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Removing Parentheses
Removing parentheses is like unwrapping a gift to reveal its contents. To simplify an expression with parentheses, you need to handle everything inside the parentheses first. Start by identifying the operation outside the parentheses. For instance, in our example, the expression was:

2 [2a + 1 - (3a - 6)].

Here, there is an outer number (2) and an inner expression (3a - 6). Let's tackle the inner expression by removing those parentheses. Always remember to apply the necessary operations that involve distributing or combining within the parentheses first.
Combining Like Terms
Combining like terms is like grouping similar items together. Terms with the same variable parts (like '2a' and '-3a') can be combined. Once the parentheses are removed, identify these terms. In our example:

2a + 1 - 3a + 6..

Here, '2a' and '-3a' are like terms, and '1' and '6' are constants. Combine them step-by-step:
  • 2a - 3a = -a.
  • 1 + 6 = 7.

Thus, simplifying the expression to -a + 7. By combining like terms, you make the expression simpler and more manageable.
Distributive Property
The distributive property is a powerful algebra tool that helps to multiply a number by a sum or difference inside parentheses. It's like distributing candies equally into different bags. The formula is:

a(b + c) = ab + ac.

In our solved example:

2[-a + 7],

distributing the '2' across both terms inside the brackets:
  • 2 * -a = -2a
  • 2 * 7 = 14.

Thus, the final simplified expression becomes: -2a + 14. By using the distributive property, you simplify and open up the expression.
Negative Sign Distribution
Distributing a negative sign can feel tricky, but it's just like distributing a regular number. When you distribute a negative sign, it changes the signs of the terms inside the parentheses. Let's look at our example:

2a + 1 - (3a - 6).

Here, the '-' before the parentheses means that both terms inside change signs:
  • -3a becomes -3a,
  • -(-6) becomes +6.

This gives us:

2a + 1 - 3a + 6. By correctly distributing the negative sign, we ensure that all terms are accurate, ready for combining and further simplification. Pay close attention to these signs, as it's a common area for mistakes!

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