Solve each equation or inequality. |5x-1| = |2x+3|

Verified step by step guidance

1. Recognize that you are dealing with an equation involving absolute values: $|5x-1|=|2x+3|$.
2. Understand that the equation $|A|=|B|$ implies two possible cases: either $A=B$ or $A=-B$.
3. Set up the first case: $5x-1=2x+3$. Solve for $x$ by isolating $x$ on one side of the equation.
4. Set up the second case: $5x-1=-(2x+3)$. Distribute the negative sign and solve for $x$ by isolating $x$ on one side of the equation.
5. Check both solutions in the original equation to ensure they satisfy $|5x-1|=|2x+3|$.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. In the context of equations, it indicates that the expressions inside the absolute value can yield two possible equations, one for each case of the sign.

Equations with Absolute Values

When solving equations involving absolute values, such as |A| = |B|, it is essential to consider both scenarios: A = B and A = -B. This leads to two separate equations that must be solved individually. The solutions from both equations must then be checked to ensure they satisfy the original equation.

Inequalities and Solution Sets

Inequalities express a relationship where one side is not necessarily equal to the other, often represented with symbols like <, >, ≤, or ≥. When solving inequalities, the solution set may include a range of values rather than discrete points. Understanding how to represent and interpret these solution sets is crucial for accurately conveying the results of an inequality.