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The distributive property is a fundamental principle in algebra that helps in simplifying expressions and solving equations. It states that multiplying a sum by a number gives the same result as multiplying each addend separately by the number and then adding the products. This property is expressed as \( a(b + c) = ab + ac \).

Let's see how the distributive property applies to our exercise. In the expression \(0.6-0.2(x+1)\), we apply this property by multiplying \(-0.2\) with both \(x\) and \(1\). This results in \(-0.2x - 0.2\).

This property makes complex expressions more manageable, allowing us to work with simpler terms that are easier to handle later on in the problem-solving process.

Equation solving is the process of finding the value of the variable that makes the equation true. It's a step-by-step approach where we simplify and manipulate the equation to isolate the variable, typically on one side of the equation.

In the exercise, the equation \(0.6-0.2(x+1)=0.4\) was solved through simplification and isolation of \(x\). First, we used the distributive property to expand \(-0.2(x+1)\) to \(-0.2x - 0.2\). Then, we combined like terms to obtain \(0.4 - 0.2x = 0.4\).

To isolate \(x\), we subtracted \(0.4\) from both sides, resulting in \(-0.2x = 0\). Finally, we divided both sides by \(-0.2\) to find \(x = 0\). This solution shows that solving equations involves multiple steps, including distribution, combining terms, and performing operations on both sides of the equation.

Combining like terms is an essential skill in simplifying algebraic expressions. Like terms are terms in an expression that have the same variable raised to the same power. By adding or subtracting these terms, we simplify the expression.

In the step \(0.6 - 0.2(x+1)\), after using the distributive property, we had \(0.6 - 0.2x - 0.2\). Here, \(0.6\) and \(-0.2\) are the like terms because they are constant terms with no variables. Adding these together gives \(0.4\), simplifying the expression to \(0.4 - 0.2x\).

Combining like terms reduces the complexity of expressions, making them easier to work with, whether you are simplifying an expression or solving an equation.