Determine whether each statement is true or false. If the...

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What's wrong with this argument? Suppose $x$ and $y$ represent two real numbers, where $x>y .$ $$\begin{aligned}2 &>1 \\2(y-x) &>1(y-x) \\2 y-2 x &>y-x \\y-2 x &>-x \\y &>x\end{aligned}$$ This is a true statement. Multiply both sides by $y-x$ Use the distributive property. Subtract $y$ from both sides. Add $2 x$ to both sides. The final inequality, $y>x,$ is impossible because we were initially given $x>y$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
What's wrong with this argument? Suppose $x$ and $y$ represent two real numbers, where $x>y .$
$$\begin{aligned}2 &>1 \\2(y-x) &>1(y-x) \\2 y-2 x &>y-x \\y-2 x &>-x \\y &>x\end{aligned}$$
This is a true statement. Multiply both sides by $y-x$ Use the distributive property. Subtract $y$ from both sides. Add $2 x$ to both sides.
The final inequality, $y>x,$ is impossible because we were initially given $x>y$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Multiplying or Dividing by a Negative Number

When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This rule is critical in maintaining the truth of the inequality when performing such operations.

Properties of Inequalities

Understanding inequalities involves knowing how various operations affect them. This includes the additive and multiplicative properties, where adding or subtracting the same quantity preserves the inequality, but multiplying or dividing by a negative number requires reversing the inequality sign.

Logical Consistency in Mathematical Arguments

A valid mathematical argument must maintain consistency with its initial assumptions and apply all operations correctly. In the context of inequalities, any incorrect application of arithmetic rules—such as failing to reverse an inequality when multiplying by a negative—can lead to conclusions that contradict the starting assumptions.

Transcript

Please give Ace some feedback