Absolute value inequalities are equations that contain the absolute value of a single variable. The absolute value of a number is the distance the number is from zero on a number line. An absolute value inequality can be used to represent a range of values, and graphing it can be a useful way of visualizing the range of solutions to the equation. Here is an overview of how to graph absolute value inequalities.
Step 1: Simplify the Equation
The first step in graphing an absolute value inequality is to simplify the equation. An absolute value equation can be written in two forms: the standard form and the piecewise form. In standard form, the equation will have the absolute value symbol followed by the variable, with an inequality symbol and a number on the other side. In piecewise form, the equation will be split into two parts, with the first part representing the case when the absolute value is greater than or equal to zero, and the second part representing the case when the absolute value is less than zero.
To simplify an absolute value equation in standard form, start by solving for the absolute value symbol. This will result in two results, one positive and one negative. To simplify an equation in piecewise form, start by solving each part separately and then combining the solutions. This will result in an equation with the inequality symbol and a number on one side, and the variable on the other.
Step 2: Graph the Equation
Once the equation has been simplified, the next step is to graph the equation. To do this, start by plotting the solutions on a number line. This will be the two points that represent the solutions to the equation. For example, if the equation is x > 4, the solutions would be 4 and infinity. Then draw a straight line between the two points, with an arrow pointing to the right if the inequality is greater than or equal to, or an arrow pointing to the left if the inequality is less than.
Once the line has been drawn, draw a shaded region on one side of the line. The side that is shaded will be determined by the inequality symbol. For example, if the inequality is greater than or equal to, the side that is shaded will be the side of the line that contains the arrow pointing to the right. This shaded region is the range of values that satisfy the equation.
Step 3: Analyze the Solution
The last step in graphing absolute value inequalities is to analyze the solutions. This can be done by looking at the graph and determining which values satisfy the equation. For example, if the inequality is x ≥ 4, then any number greater than or equal to 4 will satisfy the equation. This includes 4, 5, 6, and so on. Likewise, if the inequality is x ≤ 4, then any number less than or equal to 4 will satisfy the equation. This includes 4, 3, 2, and so on.
Conclusion
Graphing absolute value inequalities can be a useful way of visualizing the range of solutions to an equation. To graph an absolute value inequality, start by simplifying the equation, then graph the equation by plotting the solutions on a number line and drawing a shaded region on one side of the line. Finally, analyze the solution by determining which values satisfy the equation. With some practice, graphing absolute value equations can become a straightforward process.
