Question

ms ordonez/ms sparacello/ ms brea
11 a number line, drawn to scale, is shown below. the numbers 0 and 1 are
marked, as are two other numbers a and b.
step a: circle all the values that will have a positive answer
i. ( a - 1 ) iv. ( | b | )
ii. ( - b ) v. ( a + b )
iii. ( | a - b | ) vi. ( b - a )

Explanation:

Step1: Analyze \(a - 1\)

Since \(a>1\) (from the number - line), then \(a - 1>0\).

Step2: Analyze \(-b\)

Since \(b < 0\) (from the number - line), then \(-b>0\) (the opposite of a negative number is positive).

Step3: Analyze \(|a - b|\)

The absolute value of any non - zero number is positive. Since \(a
eq b\), \(|a - b|>0\).

Step4: Analyze \(|b|\)

The absolute value of any non - zero number is positive. Since \(b
eq0\), \(|b|>0\).

Step5: Analyze \(a + b\)

Let the distance between \(0\) and \(1\) be \(d\). Assume the distance between \(b\) and \(0\) is \(3d\) and the distance between \(1\) and \(a\) is \(d\). Then \(a = 2\) and \(b=-3\), \(a + b=-1<0\). In general, \(|b|>|a|\) (from the number - line: the distance of \(b\) from \(0\) is greater than the distance of \(a\) from \(1\) and \(a>1\)), so \(a + b=(|a|-|b|)\) (when \(b<0,a>0\)) and \(a + b<0\).

Step6: Analyze \(b - a\)

\(b - a=-(a - b)\). Since \(a>b\) (from the number - line), \(a - b>0\), so \(b - a<0\).

Answer:

i. \(a - 1\), ii. \(-b\), iii. \(|a - b|\), iv. \(|b|\)