Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$-[x-(4 x+2)]=2+(2 x+7)$$

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Simplify the Equation

The given equation is \[-[x-(4x+2)] = 2 + (2x+7)\].Start by simplifying the expression inside the brackets on the left side.\[x - (4x + 2) = x - 4x - 2 = -3x - 2\].Apply the negative sign: \[-(-3x - 2) = 3x + 2\].So the equation becomes \[3x + 2 = 2 + 2x + 7\].

Step 2: Combine Like Terms

Now, simplify the right side of the equation:\[2 + 2x + 7 = 2x + 9\].The equation is now: \[3x + 2 = 2x + 9\].

Step 3: Solve for x

Subtract \(2x\) from both sides to isolate \(x\):\[3x - 2x + 2 = 2x - 2x + 9\], which simplifies to\[x + 2 = 9\].Now, subtract 2 from both sides:\[x + 2 - 2 = 9 - 2\], giving\[x = 7\].

Step 4: Check the Solution Analytically

Substitute \(x = 7\) back into the original equation:\[-[7 - (4 \times 7 + 2)] = 2 + (2 \times 7 + 7)\].First simplify inside the brackets:\[4 \times 7 + 2 = 28 + 2 = 30\]\[7 - 30 = -23\]\[-(-23) = 23\].Now simplify the right side:\[2 + (2 \times 7 + 7) = 2 + 14 + 7 = 23\].Both sides are equal, confirming \(x = 7\) is correct.

Step 5: Graphical Support (Conceptual Outline)

To support the solution graphically, plot the functions \(y = -[x - (4x + 2)]\) and \(y = 2 + (2x + 7)\) on the same graph.The solution \(x = 7\) corresponds to the x-value where the graphs of these functions intersect.This graphical check confirms that the analytical solution is accurate.

Key Concepts

Graphical Representation

Understanding how a graphical representation of an equation works can make it much clearer. Essentially, when we graph the equation

• \(y = -[x - (4x + 2)]\) and
• \(y = 2 + (2x + 7)\)

on the same coordinate plane, we are looking for points where these two graphs intersect. The intersection point(s) represent the solution(s) to the equation.
For this particular problem, the solution occurs when \(x = 7\), so when we graph both equations:

• The left side of the equation transforms into \(y = 3x + 2\).
• The right side simplifies to \(y = 2x + 9\).

When plotted, these functions will intersect at the point where \(x = 7\). This graphical method visually affirms that \(x = 7\) satisfies the original equation.

Combining Like Terms

Simplifying an equation often involves combining like terms. Like terms are terms that have the same variable raised to the same power.
In the equation given, we started with a complex expression

• \(-[x - (4x + 2)] = 2 + (2x + 7)\).

To simplify, we first manage each side of the equation separately.

• The expression inside the brackets \(x - (4x + 2)\) simplifies to \(-3x - 2\).

Applying the negative sign, we get \(3x + 2\).
On the right side, combining \(2\) with \(7\) results in the expression \(2x + 9\).
By mastering the art of combining like terms, handling complex equations becomes significantly more manageable, creating streamlined equations that are easier to solve.

Checking Solutions Analytically

Once you find a solution to an equation, it's crucial to verify its accuracy, and this is done by checking solutions analytically.
For the given equation, we calculated that \(x = 7\). The next logical step is to plug this value back into the original equation:

• Substitute \(x = 7\): \[-[7 - (4 \times 7 + 2)] = 2 + (2 \times 7 + 7)\]
• Calculate inside the brackets:
  \(4 \times 7 + 2 = 30\), \(7 - 30 = -23\)
• The left side evaluates to \(-(-23) = 23\).

Now do the same operations on the right side:

• The calculation becomes \(2 + 14 + 7 = 23\).

Since both sides of the equation equal 23 when \(x = 7\), this confirms our solution is correct. Analytical checking is a valuable habit that ensures accuracy throughout mathematical problem-solving.