key term - Multiplication Property of Inequality

Definition

The multiplication property of inequality states that if an inequality is multiplied by a positive number, the inequality is preserved. However, if an inequality is multiplied by a negative number, the direction of the inequality is reversed.

5 Must Know Facts For Your Next Test

1. The multiplication property of inequality states that if an inequality is multiplied by a positive number, the inequality is preserved. For example, if $x \geq 5$, then $3x \geq 15$.
2. If an inequality is multiplied by a negative number, the direction of the inequality is reversed. For example, if $x \geq 5$, then $-2x \leq -10$.
3. The multiplication property of inequality is often used to solve linear inequalities and absolute value inequalities by isolating the variable on one side of the inequality.
4. Multiplying both sides of an inequality by the same positive or negative number is a valid step in solving the inequality, as it preserves the relationship between the two sides.
5. The multiplication property of inequality is an important concept in understanding how to manipulate and solve various types of inequalities in algebra.

Review Questions

• Explain how the multiplication property of inequality can be used to solve a linear inequality, such as $2x + 3 \geq 5$.
  • To solve the linear inequality $2x + 3 \geq 5$ using the multiplication property of inequality, we would first subtract 3 from both sides to isolate the variable term: $2x \geq 2$. Then, we would divide both sides by 2 to solve for $x$: $x \geq 1$. The multiplication property of inequality allows us to perform these steps while preserving the direction of the inequality, ensuring that the solution set includes all the values of $x$ that satisfy the original inequality.
• Describe how the multiplication property of inequality can be used to solve an absolute value inequality, such as $|x - 2| \leq 4$.
  • To solve the absolute value inequality $|x - 2| \leq 4$ using the multiplication property of inequality, we would first isolate the absolute value term by subtracting 2 from both sides: $|x - 2| \leq 4 - 2$, or $|x - 2| \leq 2$. Next, we would use the definition of absolute value to create two separate linear inequalities: $x - 2 \leq 2$ and $-(x - 2) \leq 2$. Solving these inequalities using the multiplication property, we get $x \leq 4$ and $x \geq 0$. The solution set for this absolute value inequality is the intersection of these two linear inequalities, which is the interval $[0, 4]$.
• Analyze the effect of multiplying both sides of an inequality by a negative number, and explain how this impacts the solution set.
  • When both sides of an inequality are multiplied by a negative number, the direction of the inequality is reversed. For example, if the original inequality is $x \geq 5$, and we multiply both sides by -2, the resulting inequality is $-2x \leq -10$. This means that the solution set for the new inequality is the opposite of the solution set for the original inequality. In this case, the solution set changes from $x \geq 5$ to $x \leq -5$. Understanding the effect of multiplying by a negative number is crucial when solving inequalities, as it allows you to correctly identify the solution set and ensure that the final answer satisfies the original inequality.

Related terms

Linear Inequality:

A linear inequality is an inequality that can be written in the form $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are real numbers and $x$ is a variable.

Absolute Value Inequality:

An absolute value inequality is an inequality that involves the absolute value of a variable, such as $|x| \geq a$ or $|x| \leq a$, where $a$ is a real number.

Inverse Operation: An inverse operation is an operation that undoes the effect of another operation. For example, addition and subtraction are inverse operations, as are multiplication and division.