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Graphing functions is an effective way to visually understand solutions to equations. To graph the function given in an equation, each expression on either side represents a separate function. In our exercise, you'll graph the left side: \(3(x+2)-5x\) and the right side: \(3x-4\).

Using graphing tools:

• Plot the function \(y = -2x + 6\) for the left-hand side.
• Plot the function \(y = 3x - 4\) for the right-hand side.

Observe where these two lines intersect on the graph. The x-coordinate of this intersection point is the solution to the equation. In this exercise, they intersect at \(x = 2\), which solves the original equation.

Graphing helps not only to find solutions but also to understand the behavior of functions, as you can see which function grows faster or intersects certain points on the coordinate plane.

The distributive property is a key tool in algebra that allows you to multiply a single term by multiple terms inside a set of parentheses. This helps simplify complex expressions.

In our equation \(3(x+2)-5x\), apply the distributive property as follows:

• Multiply \(3\) by \(x\) to get \(3x\).
• Multiply \(3\) by \(2\) to get \(6\).

Putting it together, you simplify the expression to \(3x + 6\).

Using the distributive property helps break down larger problems into more manageable pieces, making it easier to proceed with solving or simplifying an equation.

Combining like terms is a fundamental part of simplifying equations. By doing so, you merge terms with the same variable into a single term, which streamlines the solving process.

After using the distributive property, the expression \(3x + 6 - 5x\) needs further simplification. Look at each term:

• Terms with \(x\): Combine \(3x\) and \(-5x\) to get \(-2x\).
• Constant term: \(6\) remains unchanged.

The simplified expression is \(-2x + 6\).

By combining like terms, you reduce confusion, making it simpler to identify relationships between components of the equation, a crucial step before graphing or solving the equation.

A viewing rectangle is a section of the coordinate plane that you "zoom into", allowing you to focus on areas relevant to your functions or equations.

When graphing both \(y = -2x + 6\) and \(y = 3x - 4\), your viewing rectangle should be large enough to show the intersection point, which is crucial for finding the solution to the equation. Adjusting the viewing rectangle will change how much of the graph and what part of the graph is shown, thus impacting how clearly you can see where the lines meet.

Think of the viewing rectangle as a window into the graph. By setting the appropriate axes limits, you ensure the important features of both functions are visible on your graph. For this exercise, capture the section where \(x\) could be around the solution \(x = 2\), ensuring the intersection point won't be off the screen.