How to Graph and Solve Inequalities with Interval Notation

TL;DR

To graph an inequality on a number line, use an open circle when the sign has no underline (> or <) and a closed circle when it does (≥ or ≤), then shade right for greater-than and left for less-than. Write the answer in interval notation from left to right, using parentheses for open endpoints and infinity, and brackets for closed endpoints. The video works through examples with fractions, variables on both sides, and absolute value.

Transcript

in this video we're going to focus on graphing inequalities on number lines and also solving inequalities we're going to solve inequalities with fractions variables on both sides absolute value and we're also going to write the answer in interval notation so let's begin let's say if x is greater than 2 how can you represent this inequality on a num... Read More

Key Insights

• 😚 Graphing inequalities on number lines involves determining open or closed circles and shading in the appropriate direction based on the inequality sign.
• 🧡 The interval notation is used to represent the solutions of inequalities, indicating the range of values that satisfy the inequality.
• 😑 Absolute value inequalities can be solved by writing two equations with the positive and negative versions of the absolute value expression and finding the common solution.
• 🙃 Inequalities with fractions and variables on both sides can be solved using inverse operations to isolate the variable.

Questions & Answers

Q: How do you express a graphed inequality in interval notation?

Read the number line from left to right, from the lowest value to the highest value. Use a parenthesis for an open circle and for infinity, and a bracket for a closed circle. For example, x greater than 2 is written (2, ∞), since the open circle at 2 gets a parenthesis and infinity always takes a parenthesis.

Q: When do you use an open circle versus a closed circle on a number line?

Look at the inequality sign. If there is no underline (> or <), you use an open circle, which becomes a parenthesis in interval notation. If there is an underline (≥ or ≤), you use a closed circle, which becomes a bracket. For instance, x ≥ -1 uses a closed circle and is written [-1, ∞).

Q: Which direction do you shade when graphing an inequality?

Shade to the right when x is greater than a value, and shade to the left when x is less than a value. For example, x > 2 shades to the right of 2, while x < 4 shades to the left of 4 with an open circle at 4, giving (-∞, 4).

Q: How do you write a compound inequality like x greater than 1 and less than or equal to 4?

You have an open circle at 1 and a closed circle at 4, shading in between the two points. Because x is greater than 1 you shade right of 1, and because it is less than or equal to 4 you shade left of 4. In interval notation this is (1, 4], with a bracket at 4 for the closed circle.

Q: How do you graph an 'or' inequality such as x less than -2 or x greater than or equal to 3?

There are two separate shaded regions. Put an open circle at -2 and shade left, then a closed circle at 3 and shade right. Combine the two parts with a union symbol: (-∞, -2) ∪ [3, ∞).

Q: How do you solve an inequality like x + 4 > 5?

Isolate x by performing the opposite operation. Since 4 is added to x, subtract 4 from both sides, leaving x greater than 1. In interval notation the answer is (1, ∞), and you can then graph it on a number line with an open circle at 1 shaded to the right.

Q: How do you solve absolute value inequalities?

Write two equations, one using the positive version and one using the negative version of the absolute value expression, then find the common solution. The video covers this alongside inequalities with fractions and variables on both sides, which are solved using inverse operations to isolate the variable.

Summary & Key Takeaways

• The video explains how to graph inequalities on number lines and solve them, covering topics such as open and closed circles, shading, and interval notation.

• It provides examples of graphing and solving inequalities, including cases with fractions and variables on both sides of the inequality.

• The solutions are represented in interval notation, which shows the range of values that satisfy the inequality.