Problem 26 - Algebra 2

Question

Solve each equation. Check your solutions. \(11=|3 x+5|\)

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -\frac{16}{3}\).

Step 1: Understanding Absolute Value Equations

When solving equations involving absolute values, such as \(11 = |3x + 5|\), it's important to remember that the expression within the absolute value can be either positive or negative. Thus, we must consider two separate equations for \(3x + 5\): one where it's equal to 11 and another where it's equal to -11.

Step 2: First Case - Positive Solution

Set the expression within the absolute value equal to 11: \[3x + 5 = 11\] Subtract 5 from both sides: \[3x = 6\] Divide both sides by 3 to solve for \(x\): \[x = 2\].

Step 3: Second Case - Negative Solution

Set the expression within the absolute value equal to -11: \[3x + 5 = -11\] Subtract 5 from both sides: \[3x = -16\] Divide both sides by 3 to solve for \(x\): \[x = -rac{16}{3}\].

Step 4: Check Both Solutions

Substitute \(x = 2\) back into the original equation: \(11 = |3(2) + 5| = |6 + 5| = |11| = 11\). This is true, so \(x = 2\) is a valid solution.For \(x = -\frac{16}{3}\), substitute back into the original equation: \[11 = \left| 3\left(-\frac{16}{3}\right) + 5 \right| = \left| -16 + 5 \right| = |-11| = 11\]. This is also true, so \(x = -\frac{16}{3}\) is a valid solution as well.

Key Concepts

Solving Equations

To solve absolute value equations, like the one in this exercise, we start by understanding that an absolute value equation can yield two potential solutions. This is because the expression inside the absolute value can be equal to a positive number or its negative counterpart. Here, the equation given is \(11 = |3x + 5|\). Breaking it down, we consider two scenarios:

• \(3x + 5 = 11\)
• \(3x + 5 = -11\)

By considering both cases, we capture all possible solutions to the equation.

Two-Step Equations

Solving each of these scenarios follows the procedure for two-step equations. For the positive case, set \(3x + 5 = 11\).

• Subtract 5 from both sides to isolate the term with \(x\), resulting in \(3x = 6\).
• Next, divide both sides by 3 to solve for \(x\), giving \(x = 2\).

For the negative case, set \(3x + 5 = -11\).

• Again, subtract 5 from both sides, giving \(3x = -16\).
• Then divide by 3 to isolate \(x\), finding \(x = -\frac{16}{3}\).

This step-by-step unwinding is typical for two-step equations and helps simplify the solution process.

Checking Solutions

After solving for \(x\) in both potential cases, it's crucial to verify the solutions to ensure they satisfy the original equation. This involves substituting each solution back into the original absolute value equation.

• First check with \(x = 2\): Substitute into the equation to confirm \(11 = |3(2) + 5| = |11| = 11\), showing the solution is valid.
• Next, verify \(x = -\frac{16}{3}\): Substitute into the equation to see \(11 = \left| 3\left(-\frac{16}{3}\right) + 5 \right| = |-11| = 11\), which confirms validity as well.

Checking solutions provides confidence that both values for \(x\) are indeed correct and satisfy the equation.