Question

tuesday
#5 compare the numbers below using <, > or =
1.76 ____ |-1.69|
#6 evaluate the expression below. simplify your answer.
\\(\frac{3}{4} + 2\frac{1}{2}\\)
#7 evaluate the expression below. simplify your answer.
\\(1\frac{2}{5} \div 4\\)

Explanation:

#5: Compare the numbers

Step1: Find absolute value

The absolute value of a number \( x \) is \( |x|=

$$\begin{cases}x, & x\geq0\\ -x, & x < 0\end{cases}$$

\). For \( |- 1.89| \), since \( - 1.89<0 \), \( |-1.89|=-(-1.89) = 1.89 \)

Step2: Compare 1.76 and 1.89

We know that when comparing two positive numbers \( a \) and \( b \), if \( a < b \), then \( a \) is less than \( b \). Here \( 1.76\) and \( 1.89\) are positive numbers and \( 1.76<1.89 \)

Step1: Convert mixed number to improper fraction

A mixed number \( a\frac{b}{c}=\frac{ac + b}{c} \). So \( 2\frac{1}{2}=\frac{2\times2 + 1}{2}=\frac{4 + 1}{2}=\frac{5}{2} \)

Step2: Find a common denominator

The denominators are 4 and 2. The least common denominator of 4 and 2 is 4. Convert \( \frac{5}{2} \) to a fraction with denominator 4: \( \frac{5}{2}=\frac{5\times2}{2\times2}=\frac{10}{4} \)

Step3: Add the fractions

\( \frac{3}{4}+\frac{10}{4}=\frac{3 + 10}{4}=\frac{13}{4} \)

Step4: Convert back to mixed number (optional)

\( \frac{13}{4}=3\frac{1}{4} \)

Step1: Convert mixed number to improper fraction

\( 1\frac{2}{5}=\frac{1\times5+2}{5}=\frac{5 + 2}{5}=\frac{7}{5} \)

Step2: Divide by 4 (multiply by reciprocal)

Dividing by a number \( n \) is the same as multiplying by \( \frac{1}{n} \). So \( \frac{7}{5}\div4=\frac{7}{5}\times\frac{1}{4}=\frac{7\times1}{5\times4}=\frac{7}{20} \)

Answer:

\( 1.76<|-1.89| \)

#6: Evaluate \( \frac{3}{4}+2\frac{1}{2} \)