Problem 83 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ |7 x+12|=|x-6| $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{3}{4}\).

Step 1: Set Up Equations

When dealing with an equation that involves absolute values, we need to consider both the positive and negative scenarios of the expressions involved. Hence, for the equation \(|7x + 12| = |x - 6|\), we set up two separate equations: 1. \(7x + 12 = x - 6\)2. \(7x + 12 = -(x - 6)\).

Step 2: Solve First Equation

Solve the equation from the first scenario, \(7x + 12 = x - 6\). Begin by subtracting \(x\) from both sides: \[7x - x + 12 = -6\] This simplifies to \[6x + 12 = -6\]. Subtract 12 from both sides: \[6x = -18\]. Finally, divide both sides by 6 to solve for \(x\): \[x = -3\].

Step 3: Solve Second Equation

Solve the equation from the second scenario, \(7x + 12 = -(x - 6)\). Distribute the negative sign on the right side: \[7x + 12 = -x + 6\]. Add \(x\) to both sides: \[7x + x + 12 = 6\] This simplifies to \[8x + 12 = 6\]. Subtract 12 from both sides: \[8x = -6\]. Finally, divide both sides by 8 to solve for \(x\): \[x = -\frac{3}{4}\].

Step 4: Verify Solutions

For each solution, verify that it satisfies the original equation. - Substitute \(x = -3\) into \(|7x + 12| = |x - 6|\): \(|7(-3) + 12| = |-3 - 6|\) simplifies to \( |-9 + 12| = |-9| \) which is \(|3| = 9|\). This does not hold. - Substitute \(x = -\frac{3}{4}\) into \(|7x + 12| = |x - 6|\): \(|7(-\frac{3}{4}) + 12| = |-\frac{3}{4} - 6|\) simplifies to \(|-\frac{21}{4} + \frac{48}{4}| = |-(\frac{3}{4} + \frac{24}{4})| = |9.75|\) = |\frac{75}{4}|\, which holds true.

Step 5: Conclusions

The only solution that satisfies the original equation is \(x = -\frac{3}{4}\). Since the problem does not involve inequalities, there is no graph or interval notation necessary in this case.

Key Concepts

Solving Equations

When solving equations that involve absolute values, it's crucial to understand that absolute values represent the distance of a number from zero on a number line. This means that the expression within the absolute value can either be positive or negative.

In the given exercise, \(|7x + 12| = |x - 6|\), two scenarios must be considered:

• First scenario: \(7x + 12 = x - 6\), which considers when both expressions inside the absolute values are directly equal.
• Second scenario: \(7x + 12 = -(x - 6)\), which considers when one expression equals the negative of the other.

By solving these two scenarios individually, you can find potential solutions to the equation. This process involves typical algebraic methods like adding, subtracting, and dividing to isolate the variable.

If done correctly, you would find two possible solutions: \(x = -3\) from the first scenario, and \(x = -\frac{3}{4}\) from the second scenario.

Absolute Value Properties

The absolute value properties are pivotal when approaching problems involving absolute value equations. An absolute value function is defined as \(|x| = x\) if \(x \ge 0\) and \(|x| = -x\) if \(x < 0\).

Understanding these properties allows you to set up equations corresponding to the positive and negative possibilities of the expression inside the absolute value. For example, with the equation \(|7x + 12| = |x - 6|\), these properties help you explore both conditions:

• When \(7x + 12\) is equal to \(x - 6\).
• And when \(7x + 12\) equals the negative, \(-(x - 6)\).

By applying these properties correctly, you'll be prepared to translate absolute value equations into more solvable linear equations, making it easier to evaluate the potential solutions.

Algebraic Verification

After solving the equations from each scenario, it's crucial to verify each potential solution to see if it satisfies the original equation. Verification ensures that no errors were made during the calculation and that each solution is valid.

To verify the solutions for \(|7x + 12| = |x - 6|\):

• Substitute \(x = -3\):\(|7(-3) + 12|\) does not equal \(|-3 - 6|\), since \(|3| = 9|\), this solution does not hold.
• Substitute \(x = -\frac{3}{4}\): This simplifies correctly, showing it satisfies the original equation.

This process ensures that you can confidently conclude that \(x = -\frac{3}{4}\) is the correct solution because it results in equivalent expressions on both sides of the equation. Verification is a key step to confirm that the solution not only satisfies the algebra involved but also fits within the initial conditions set by the absolute values.