In Exercises 91–100, find all values of x satisfying the given conditions. y = |2 - 3x| and y = 13

Verified step by step guidance

1. Start by setting the two expressions for y equal to each other: |2 - 3x| = 13.
2. Recall that the absolute value equation |A| = B can be split into two separate equations: A = B and A = -B.
3. Apply this to the equation |2 - 3x| = 13, resulting in two cases: 2 - 3x = 13 and 2 - 3x = -13.
4. Solve the first equation, 2 - 3x = 13, by isolating x. Subtract 2 from both sides to get -3x = 11, then divide by -3.
5. Solve the second equation, 2 - 3x = -13, by isolating x. Subtract 2 from both sides to get -3x = -15, then divide by -3.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, represents the distance of x from zero on the number line, always yielding a non-negative result. In the context of the equation y = |2 - 3x|, it indicates that the expression inside the absolute value can yield two scenarios: one where the expression is positive and another where it is negative, leading to two separate equations to solve.

Setting Equations Equal

To find the values of x that satisfy both equations y = |2 - 3x| and y = 13, we set them equal to each other. This means we will solve the equation |2 - 3x| = 13, which will involve breaking it down into two cases: 2 - 3x = 13 and 2 - 3x = -13. Each case will yield potential solutions for x.

Solving Linear Equations

Solving linear equations involves isolating the variable (in this case, x) to find its value. This process typically includes performing operations such as addition, subtraction, multiplication, or division on both sides of the equation. After determining the values of x from the cases derived from the absolute value equation, it is essential to verify that these solutions satisfy the original conditions.