<\/span><\/h2>\nTo solve an absolute value inequality algebraically, follow these steps:<\/p>\n
1. Identify the inequality sign:<\/strong> Determine whether the inequality is greater than (>) or less than (<).<\/p>\n2. Isolate the absolute value:<\/strong> Ensure that the expression inside the absolute value is by itself on one side of the inequality.<\/p>\n3. Set up two inequalities:<\/strong> Considering the nature of absolute value, set up two inequalities, one positive and one negative, removing the absolute value symbol.<\/p>\n4. Solve for the variable:<\/strong> Solve both inequalities separately to find the possible solutions. In other words, solve the equation twice while changing the inequality symbol for the second equation.<\/p>\n5. Combine the solutions:<\/strong> Combine the solutions from both inequalities, using the appropriate inequality sign to represent the combination of the solutions.<\/p>\nLet’s illustrate these steps through an example:<\/p>\n
Example:<\/strong> Solve the absolute value inequality |2x – 1| \u2264 5.<\/p>\n1. The inequality sign in this case is (\u2264), indicating that the expression inside the absolute value must be less than or equal to 5.<\/p>\n
2. To isolate the absolute value, rewrite the inequality as 2x – 1 \u2264 5.<\/p>\n
3. Set up two inequalities, one positive and one negative:<\/p>\n
Positive: 2x – 1 \u2264 5
\n Negative: -(2x – 1) \u2264 5<\/p>\n
4. Solve for the variable:<\/p>\n
Positive: 2x \u2264 6 \u27f9 x \u2264 3
\n Negative: -2x + 1 \u2264 5 \u27f9 -2x \u2264 4 \u27f9 x \u2265 -2<\/p>\n
5. Combine the solutions using the appropriate inequality signs:<\/p>\n
The final solution is -2 \u2264 x \u2264 3.<\/p>\n