Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$-(8+3 x)+5=2 x+3$$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{6}{5}\).

Step 1: Simplify the equation

First, distribute the negative sign on the left-hand side of the equation. This gives \[-(8+3x)= -8 - 3x.\]So the original equation \[-(8+3x)+5 = 2x + 3\]becomes \[-8 - 3x + 5 = 2x + 3.\]

Step 2: Combine like terms

Next, combine like terms on the left-hand side:\[-8 + 5 = -3.\]This simplifies the equation to:\[-3x - 3 = 2x + 3.\]

Step 3: Bring variable terms to one side

Move all terms involving \(x\) to one side of the equation by adding \(3x\) to both sides:\[-3 = 5x + 3.\]

Step 4: Isolate the variable

Now, isolate \(x\) by subtracting 3 from both sides of the equation:\[-3 - 3 = 5x\]. This simplifies to:\[-6 = 5x.\]

Step 5: Solve for the variable

Divide both sides by 5 to solve for \(x\):\[x = \frac{-6}{5}.\]

Step 6: Verify the solution

Substitute \(x = -\frac{6}{5}\) back into the original equation to verify:\[-(8+3(-\frac{6}{5}))+5 = 2(-\frac{6}{5})+3.\]Simplify both sides:\[-(8 - \frac{18}{5}) + 5 = -\frac{12}{5} + 3.\]Calculate numerically to check equality:\[-(\frac{40}{5} - \frac{18}{5}) + \frac{25}{5} = \frac{15}{5} - \frac{12}{5}.\]This simplifies to \[-\frac{22}{5} + \frac{25}{5} = \frac{3}{5}.\]Both sides are equal, confirming the solution is correct.

Step 7: Graphical support

Graph the functions \(f(x) = -(8+3x) + 5\) and \(g(x) = 2x + 3\).They should intersect at \(x = -\frac{6}{5}\), which supports our analytical solution visually.

Key Concepts

Equation simplification

Simplifying an equation is like cleaning a messy room. It makes everything more clear and manageable. You do this by reducing the equation to its simplest form. In our example exercise, you start by distributing the negative sign on the left-hand side. This changes \[-(8 + 3x)\] into \[-8 - 3x\]. Next, you continue simplifying by combining any like terms on the same side of the equation. These like terms are ones that share the same variable or are constants. So, \[-8 + 5 = -3\] results in a simpler expression: \[-3x - 3\].

• Distribute: Apply multiplication across parentheses.
• Combine like terms: Add or subtract similar terms (constants or those with the same variable).

Simplifying makes it easier to see the path to solving for the variable, and it lays a solid foundation for the next steps you’ll take in solving the equation.

Variable isolation

Variable isolation is an essential step in solving equations. Imagine setting aside one apple from a group of different fruits, making it easier to focus solely on that apple. We aim to isolate the variable so it stands alone on one side of the equation.To do this, start by moving all variable terms to one side. In the exercise, \(-3x - 3 = 2x + 3\) requires you to add \(3x\) to both sides, yielding \(-3 = 5x + 3\). This movement of terms is key; it transforms the equation into a format where you can more easily solve for the variable \(x\).

• Rearrange terms: Move variables to one side and constants to the other.
• Use inverse operations: Counteract addition with subtraction or vice versa to simplify.

The ultimate goal with variable isolation is to reach a point where the variable is by itself, allowing you to find its value straightforwardly.

Verification of solution

Verification is the process that checks if our solution truly solves the given equation. If you solve a puzzle, you naturally check if all pieces fit to ensure that it’s completed correctly. This is the same concept in algebra.To verify the solution of \(x = -\frac{6}{5}\), substitute this value back into the original equation. You perform this check to prove that both sides of the equation are equal. In our case, substituting yields an equivalent expression on either side, confirming the accuracy of the solution.

• Substitute the solution back into the original equation.
• Ensure both sides of the equation produce the same value when simplified.

Verification not only builds your confidence in the solution but also acts as a safety net, ensuring that all steps taken were done correctly.