Question

using this number line, which of the following orderings is correct? 0 0.35 0.5 0.81 1 |0.35| < -|0.81| < 0 < 0.5 < |-0.81| < 1 |-0.81| < 0 < |0.35| < 0.5 < -|0.81| < 1 0 < |-0.35| < 0.5 < -|-0.81| < 1 -|0.81| < 0 < |0.35| < 0.5 < |-0.81| < 1

Explanation:

Step1: Calculate absolute values and negatives

First, recall that the absolute value of a number \(|x|\) is its distance from 0, so it's non - negative. For a positive number \(a\), \(|a| = a\), and for a negative number \( - a\) (where \(a>0\)), \(|-a|=a\), and \(-|a|=-a\).

• Calculate \(|0.35|\): Since \(0.35>0\), \(|0.35| = 0.35\).
• Calculate \(-|0.81|\): Since \(|0.81| = 0.81\), then \(-|0.81|=- 0.81\).
• Calculate \(|-0.81|\): Since \(-0.81\) is negative, \(|-0.81| = 0.81\).
• Calculate \(|-0.35|\): Since \(-0.35\) is negative, \(|-0.35| = 0.35\).
• Calculate \(-|-0.81|\): First, \(|-0.81| = 0.81\), then \(-|-0.81|=-0.81\)

Step2: Analyze each option

• Option 1: \(|0.35| = 0.35\), \(-|0.81|=-0.81\), \(|-0.81| = 0.81\). The order \(0.35<-0.81 < 0<0.5 < 0.81<1\) is wrong because \(-0.81<0\) and \(0.35>0\), so \( - 0.81<0<0.35\) is the correct order among these values, not \(0.35<-0.81\).
• Option 2: \(|-0.81| = 0.81\), \(|0.35| = 0.35\), \(-|0.81|=-0.81\). The order \(0.81<0<0.35<0.5<-0.81 < 1\) is wrong because \(0.81>0\), \(0.35>0\) and \(-0.81<0\), so \( - 0.81<0<0.35<0.81\) is the correct order among these values, not \(0.81<0\).
• Option 3: \(|-0.35| = 0.35\), \(-|-0.81|=-0.81\). The order \(0 < 0.35<0.5<-0.81<1\) is wrong because \(-0.81<0\), so \(-0.81<0\) and not \(0.5<-0.81\).
• Option 4: \(-|0.81|=-0.81\), \(|0.35| = 0.35\), \(|-0.81| = 0.81\). Let's check the order: \(-0.81<0\) (because negative numbers are less than 0), \(0<0.35\) (because \(0.35>0\)), \(0.35<0.5\) (from the number line \(0.35 < 0.5\)), \(0.5<0.81\) (from the number line \(0.5<0.81\)), \(0.81<1\) (from the number line \(0.81 < 1\)). So the order \(-|0.81| < 0<|0.35|<0.5<|-0.81|<1\) is correct.

Answer:

\(-|0.81| < 0<|0.35|<0.5<|-0.81|<1\) (the fourth option)