How to Solve Absolute Value Equations with Inequalities

TL;DR

To solve absolute value equations with inequalities, write two equations: one for the positive and one for the negative case. Graph the solutions on a number line using open circles for strict inequalities, and represent the solution in interval notation with parentheses to indicate that endpoints are not included.

Transcript

in this video we're going to focus on solving absolute value equations that contain inequalities we're going to graph it on a number line and also represent the solution using interval notation so let's start with the first one let's say if you get an equation that looks like this the absolute value of x plus three is greater than four how can we s... Read More

Key Insights

• ❎ Two equations (positive and negative) are necessary to solve absolute value equations with inequalities.
• 😚 Graphing the solutions on a number line helps visualize the shaded regions and determine open or closed circles.
• ♾️ Interval notation represents the solution using negative infinity, positive infinity, and parentheses for open circles.
• ✖️ When there are fractions in the inequality, multiply by the least common multiple of the denominators to clear the fractions.

Questions & Answers

Q: How do you solve absolute value equations with inequalities?

To solve, write two equations (positive and negative) and solve for x. Graph the solutions and represent them using interval notation.

Q: What does it mean to have an open circle in the graph?

An open circle indicates that the value is greater than or less than, but not equal to. It is represented by an open circle on the number line.

Q: How do you represent the solution using interval notation?

Interval notation uses negative infinity, positive infinity, and parentheses for open circles. The solution is represented as a union of ranges.

Q: What happens when you have fractions in the inequality?

Multiply the equation by the least common multiple of the denominators to eliminate fractions. Then, solve for x following the same steps as before.

Q: Are there any special cases when solving absolute value equations with inequalities?

Yes, when the inequality is less than or equal to a negative number, there is no solution. Be careful not to overlook this fact during the solving process.

Summary & Key Takeaways

• Absolute value equations with inequalities require writing two equations, one with a positive sign and one with a negative sign, to solve for x.

• Graphing the solutions on a number line helps determine the shaded region, with open circles for "greater than" and "less than" values.

• Representing the solution in interval notation involves using negative infinity and positive infinity, with parentheses for open circles.