key term - Addition Property of Inequality

Definition

The addition property of inequality states that if an inequality is true, adding or subtracting the same quantity to both sides of the inequality will result in a new inequality that is also true. This property allows for the manipulation of inequalities in order to solve them or perform other algebraic operations.

5 Must Know Facts For Your Next Test

1. The addition property of inequality states that if $a \geq b$, then $a + c \geq b + c$, where $a$, $b$, and $c$ are real numbers.
2. This property allows you to add or subtract the same quantity to both sides of an inequality without changing the direction of the inequality.
3. The addition property of inequality is crucial in solving linear inequalities and absolute value inequalities by isolating the variable on one side of the inequality.
4. When solving inequalities, you can use the addition property to eliminate constant terms on both sides of the inequality, making it easier to find the solution set.
5. The addition property of inequality also applies to the subtraction of the same quantity from both sides of an inequality, as subtracting a number is equivalent to adding the additive inverse of that number.

Review Questions

• Explain how the addition property of inequality can be used to solve a linear inequality.
  • To solve a linear inequality using the addition property, you would first isolate the variable term on one side of the inequality by adding or subtracting the same quantity from both sides. For example, to solve the inequality $2x + 5 \geq 7$, you would subtract 5 from both sides, resulting in $2x \geq 2$. Then, you would divide both sides by 2 to isolate the variable $x$, yielding $x \geq 1$. The addition property ensures that the new inequality, $x \geq 1$, is also true.
• Describe how the addition property of inequality can be applied to solve an absolute value inequality.
  • When solving an absolute value inequality, the addition property of inequality can be used to eliminate the constant terms on both sides of the inequality. For instance, to solve the inequality $|x - 3| \leq 5$, you would first add 3 to both sides, resulting in $|x| \leq 8$. Then, you would consider the two possible cases: $x \leq 8$ and $x \geq -8$. The addition property ensures that these new inequalities are also true, and the solution set is the intersection of the two cases, which is $-8 \leq x \leq 8$.
• Analyze how the addition property of inequality can be used to demonstrate the relationship between linear inequalities and absolute value inequalities.
  • The addition property of inequality can be used to show the connection between linear inequalities and absolute value inequalities. For example, the inequality $|x - 3| \leq 5$ can be rewritten as $-5 \leq x - 3 \leq 5$ by using the definition of absolute value. Applying the addition property, this is equivalent to $-2 \leq x \leq 8$. This demonstrates that an absolute value inequality can be transformed into a system of linear inequalities, and vice versa, by strategically adding or subtracting the same quantity from both sides. This relationship is crucial in understanding the solution sets of these types of inequalities.

Related terms

Inequality:

A mathematical statement that compares two expressions using symbols such as >, <, ≥, or ≤, indicating that one value is greater than, less than, greater than or equal to, or less than or equal to the other value.

Linear Inequality:

An inequality that can be written in the form $ax + b \geq c$ or $ax + b > c$, where $a$, $b$, and $c$ are constants and $x$ is a variable.

Absolute Value Inequality:

An inequality that involves the absolute value of a variable, such as $|x - 3| < 5$ or $|2x + 1| \geq 4$.