Solve each equation or inequality. |6 - 2x | + 1 = 3

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 1 from both sides of the equation: $|6-2x|+1-1=3-1$.
2. This simplifies to $|6-2x|=2$.
3. To solve the absolute value equation, set up two separate equations: $6-2x=2$ and $6-2x=-2$.
4. Solve the first equation: $6-2x=2$. Subtract 6 from both sides to get $-2x=-4$, then divide by -2 to find $x$.
5. Solve the second equation: $6-2x=-2$. Subtract 6 from both sides to get $-2x=-8$, then divide by -2 to find $x$.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. Understanding absolute value is crucial for solving equations and inequalities that involve expressions within these bars.

Linear Equations

A linear equation is an equation of the first degree, meaning it involves only linear terms and can be expressed in the form ax + b = c, where a, b, and c are constants. Solving linear equations involves isolating the variable on one side of the equation. In the context of the given problem, recognizing how to manipulate the equation to isolate x is essential.

Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like <, >, ≤, or ≥. When solving inequalities, it is important to consider the direction of the inequality sign, especially when multiplying or dividing by negative numbers. In the context of the problem, understanding how to interpret and solve inequalities is key to finding the solution set.