Solve the equations containing absolute value in Exercises 94–95. |2x+1| = 7

Verified step by step guidance

1. Step 1: The absolute value equation |2x+1| = 7 can be rewritten as two separate equations: 2x+1 = 7 and 2x+1 = -7. This is because the absolute value of a number is always positive, so the expression inside the absolute value can be either positive or negative.
2. Step 2: Solve the first equation 2x+1 = 7 for x. To do this, subtract 1 from both sides of the equation to isolate the term with x, then divide by 2 to solve for x.
3. Step 3: Solve the second equation 2x+1 = -7 for x. Similarly, subtract 1 from both sides of the equation to isolate the term with x, then divide by 2 to solve for x.
4. Step 4: The solutions to the original equation |2x+1| = 7 are the solutions to both equations from steps 2 and 3.
5. Step 5: Always check your solutions by substituting them back into the original equation to make sure they are correct.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. For example, |3| = 3 and |-3| = 3, indicating that both 3 and -3 are three units away from zero.

Equations with Absolute Value

When solving equations that involve absolute values, it is essential to consider both the positive and negative scenarios. For the equation |2x + 1| = 7, this means setting up two separate equations: 2x + 1 = 7 and 2x + 1 = -7. Each equation must be solved independently to find all possible solutions.

Solution Sets

A solution set is the collection of all values that satisfy a given equation. In the context of absolute value equations, it is important to check each potential solution to ensure it meets the original equation. For the example |2x + 1| = 7, the solutions must be verified to confirm they are valid.