Question Solve the equation $|2x - 1| = 18$ and find the solutions.

Studdy Solution

STEP 1

Assumptions
1. We are given the absolute value equation $|2x - 1| = 18$ .
2. We need to find all values of $x$ that satisfy the equation.

STEP 2

Understand the definition of absolute value. The absolute value of a number is its distance from zero on the number line, without considering direction. The equation $|2x - 1| = 18$ means that the expression $2x - 1$ is 18 units away from zero.

STEP 3

Set up two separate equations to account for the two possible scenarios due to the absolute value:
1. The expression inside the absolute value is positive or zero, which gives us $2x - 1 = 18$ .
2. The expression inside the absolute value is negative, which gives us $2x - 1 = -18$ .

STEP 4

Solve the first equation where the expression inside the absolute value is non-negative.
$2x - 1 = 18$

STEP 5

Add 1 to both sides of the equation to isolate the term with $x$ on one side.
$2x - 1 + 1 = 18 + 1$

STEP 6

Simplify both sides of the equation.
$2x = 19$

STEP 7

Divide both sides of the equation by 2 to solve for $x$ .
$x = \frac{19}{2}$

STEP 8

Simplify the fraction to find the first solution.
$x = 9.5$

STEP 9

Now, solve the second equation where the expression inside the absolute value is negative.
$2x - 1 = -18$

STEP 10

Add 1 to both sides of the equation to isolate the term with $x$ on one side.
$2x - 1 + 1 = -18 + 1$

STEP 11

Simplify both sides of the equation.
$2x = -17$

STEP 12

Divide both sides of the equation by 2 to solve for $x$ .
$x = \frac{-17}{2}$

STEP 13

Simplify the fraction to find the second solution.
$x = -8.5$
The solutions to the equation $|2x - 1| = 18$ are $x = 9.5$ and $x = -8.5$ .

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