To determine which of the given statements are true if [tex]$x = -2$[/tex] and [tex]$y = 8$[/tex], we need to evaluate the absolute values of [tex]$x$[/tex] and [tex]$y$[/tex] and then compare them according to the conditions provided.
1. Calculate the absolute values:
- The absolute value of [tex]$x$[/tex] is [tex]$|x| = |-2| = 2$[/tex]
- The absolute value of [tex]$y$[/tex] is [tex]$|y| = |8| = 8$[/tex]
2. Evaluate each statement:
- [tex]$|x| \leq |y|$[/tex]: This statement means the absolute value of [tex]$x$[/tex] is less than or equal to the absolute value of [tex]$y$[/tex]. Since [tex]$|x| = 2$[/tex] and [tex]$|y| = 8$[/tex], [tex]$2 \leq 8$[/tex] is true.
- [tex]$|y| \geq |x|$[/tex]: This statement means the absolute value of [tex]$y$[/tex] is greater than or equal to the absolute value of [tex]$x$[/tex]. Since [tex]$|y| = 8$[/tex] and [tex]$|x| = 2$[/tex], [tex]$8 \geq 2$[/tex] is true.
- [tex]$|x| > |y|$[/tex]: This statement means the absolute value of [tex]$x$[/tex] is greater than the absolute value of [tex]$y$[/tex]. Since [tex]$|x| = 2$[/tex] and [tex]$|y| = 8$[/tex], [tex]$2 > 8$[/tex] is false.
- [tex]$|y| < |x|$[/tex]: This statement means the absolute value of [tex]$y$[/tex] is less than the absolute value of [tex]$x$[/tex]. Since [tex]$|y| = 8$[/tex] and [tex]$|x| = 2$[/tex], [tex]$8 < 2$[/tex] is false.
Based on the evaluations:
- The first statement, [tex]$|x| \leq |y|$[/tex], is true.
- The second statement, [tex]$|y| \geq |x|$[/tex], is true.
- The third statement, [tex]$|x| > |y|$[/tex], is false.
- The fourth statement, [tex]$|y| < |x|$[/tex], is false.
Therefore, the true statements are:
- [tex]$|x| \leq |y|$[/tex]
- [tex]$|y| \geq |x|$[/tex]
