I. Intro
A relationship between two expressions or values that are not equal to each other is called inequality. It is used most often to compare two numbers by their size
The symbols used in inequalities are:
> Greater than
< Less than
≥ Greater than or equal to
≤ Less than or equal to
≠ Not equal to
Writing Solution Set In Interval Notation
While writing the solution of an inequality in the interval notation, we have to keep the following things in mind. •If the endpoint is included (i.e., in case of ≤ or ≥) use the closed brackets '[' or ']'
•If the endpoint is not included (i.e., in case of < or >), use the open brackets '(' or ')'
•Use always open bracket at either ∞ or -∞.
•If the endpoint is not included (i.e., in case of < or >), use the open brackets '(' or ')'
•Use always open bracket at either ∞ or -∞.
Inequalities On A Number Line
While graphing inequalities, we have to keep the following things in mind.•If the endpoint is included (i.e., in case of ≤ or ≥) use a closed circle.
•If the endpoint is NOT included (i.e., in case of < or >), use an open circle.
•If the endpoint is NOT included (i.e., in case of < or >), use an open circle.
•Draw a line from the endpoint that extends to the right side if the variable is greater than the number.
•Draw a line from the endpoint that extends to the left side if the variable is lesser than the number.
•Draw a line from the endpoint that extends to the left side if the variable is lesser than the number.
The inequality 𝑥>−1 would be read, “𝑥 is greater than negative one.” Notice there are many values that would make this statement true. For example, 𝑥=2 or 𝑥=5 or 𝑥=10. In fact, any number to the right of -1 on the number line would be a solution to this inequality. We can describe the solution set as an interval of real numbers. In this case, the interval (−1,∞)
II. Solving One-Step Inequalities
Solving linear inequalities is similar to solving linear equations. In fact, all the properties that we used to solve linear equations also hold for linear inequalities.
−1 × 3 < − 1 × 5
−3 ≮ −5
The same inequality would result if we divided both sides by -1. So, when we multiply or divide both sides of an inequality by a negative number, we need to switch the direction of the inequality.
Solving linear inequalities works the exact same way as solving linear equations with this one exception.
The same inequality would result if we divided both sides by -1. So, when we multiply or divide both sides of an inequality by a negative number, we need to switch the direction of the inequality.
Solving linear inequalities works the exact same way as solving linear equations with this one exception.
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