Problem 15
Question
Replace the symbol \(*\) by \(<,>,\) or \(=\) to make the statement true. \(-\frac{2}{7} *-0.285714\).
Step-by-Step Solution
Verified Answer
The equation '-2/7 * -0.285714' becomes true when '*' is replaced by the '=' symbol, yielding '-2/7 = -0.285714'.
1Step 1: Analyze the numbers
Check the two values in the exercise. '-2/7' is a fraction and '-0.285714' is a decimal.
2Step 2: Convert Fractions to Decimals
Convert '-2/7' to a decimal. The result of this is '-0.285714' as 2 divided by 7 gives '0.285714' and the negative sign is carried over.
3Step 3: Compare the values
Compare '-0.285714' (the decimal equivalent of '-2/7') and '-0.285714'. You can see that they are equal.
4Step 4: Replace the symbol
Replace the symbol '*' with ' = ' to make the equation correct. The final equation becomes '-2/7 = -0.285714'.
Key Concepts
Comparing valuesFraction to decimal conversionEquality in mathematics
Comparing values
When comparing two numbers, the first step is to ensure they're in the same format, either both as fractions or both as decimals. This makes it easier to directly compare their values.
- If both numbers have the same sign, compare their absolute values. A number with a larger absolute value is greater if positive, or smaller if negative. - If numbers have different signs, the number with a positive sign is greater than the negative number.
In our case, since both numbers are negative, we look at their absolute values. The absolute value of \(-\frac{2}{7}\) is \(\frac{2}{7}\) and for \(-0.285714\) it is \(0.285714\). Since they are equal, \(-\frac{2}{7}\) and \(-0.285714\) are equal.
- If both numbers have the same sign, compare their absolute values. A number with a larger absolute value is greater if positive, or smaller if negative. - If numbers have different signs, the number with a positive sign is greater than the negative number.
In our case, since both numbers are negative, we look at their absolute values. The absolute value of \(-\frac{2}{7}\) is \(\frac{2}{7}\) and for \(-0.285714\) it is \(0.285714\). Since they are equal, \(-\frac{2}{7}\) and \(-0.285714\) are equal.
Fraction to decimal conversion
Converting fractions to decimals is a key skill for simplifying comparisons and calculations.
- Divide the numerator by the denominator to change a fraction into a decimal. - Carry the negative sign if the fraction is negative.
For example, let's convert \(-\frac{2}{7}\) to a decimal. We perform the division: \(2 \div 7\), resulting in a repeating decimal \(0.285714\). As the fraction is negative, the decimal becomes \(-0.285714\).
You should be able to recognize that some decimals are repeating and may need to round or truncate them for practical use.
- Divide the numerator by the denominator to change a fraction into a decimal. - Carry the negative sign if the fraction is negative.
For example, let's convert \(-\frac{2}{7}\) to a decimal. We perform the division: \(2 \div 7\), resulting in a repeating decimal \(0.285714\). As the fraction is negative, the decimal becomes \(-0.285714\).
You should be able to recognize that some decimals are repeating and may need to round or truncate them for practical use.
Equality in mathematics
Equality in mathematics is a principle that states two expressions represent the same thing. To express equality, we use the symbol "=".
- For equality, both expressions must have the same value.- Mathematically, this means there is no difference between the two in value, regardless of their expression form.
In our equation \(-\frac{2}{7} = -0.285714\), equality is shown because both expressions evaluate to the same decimal value, \(-0.285714\). Understanding equality helps ensure accurate results in solving equations and verifying solutions.
- For equality, both expressions must have the same value.- Mathematically, this means there is no difference between the two in value, regardless of their expression form.
In our equation \(-\frac{2}{7} = -0.285714\), equality is shown because both expressions evaluate to the same decimal value, \(-0.285714\). Understanding equality helps ensure accurate results in solving equations and verifying solutions.
Other exercises in this chapter
Problem 14
Replace the symbol \(*\) by \(,\) or \(=\) to make the statement true. \(4 * \sqrt{16}\).
View solution Problem 15
Solve the inequality. Express the solution as an interval or as the union of intervals. Mark the solution on a number line. $$x^{2}-x-6 \geq 0$$
View solution Problem 15
Determine the domain of the function and sketch the graph. $$f(x)=\frac{1}{x^{2}-4}$$.
View solution Problem 15
Solve the inequality and mark the solution set on a number line. $$x^{3}-2 x^{2}+x \geq 0$$.
View solution