Example Question - approximate value
Here are examples of questions we've helped users solve.
Solving Inequality with Square Root Approximation
The image shows an inequality with a square root of 139, and you need to determine what numbers could go into the blank boxes so that the inequality is true. Firstly, let's find out the approximate value of √139 since it's not a perfect square. To get an idea of where it lies, you can compare it to perfect squares nearby. For instance: - √121 = 11 (since 11^2 = 121) - √144 = 12 (since 12^2 = 144) Since 139 is between 121 and 144, √139 will be between 11 and 12. Calculating the exact decimal would give you a better approximation: √139 ≈ 11.789... This means that we need to find integers that are immediately less than and greater than 11.789. The integer immediately less than 11.789 is 11, and the integer immediately greater than 11.789 is 12. Therefore, to satisfy the inequality, the numbers on the blank boxes should be 11 and 12 in that order: 11 < √139 < 12
Solving Exponential Equation for x
The equation provided is: \[ e^{4 - 7x} + 11 = 20 \] To solve for \( x \), follow these steps: 1. Subtract 11 from both sides of the equation: \[ e^{4 - 7x} = 9 \] 2. Take the natural logarithm (ln) of both sides: \[ \ln(e^{4 - 7x}) = \ln(9) \] 3. Because the natural logarithm and the exponential function are inverse operations, \( \ln(e^{y}) = y \). Hence: \[ 4 - 7x = \ln(9) \] 4. To isolate \( x \), subtract 4 from both sides: \[ -7x = \ln(9) - 4 \] 5. Finally, divide both sides by -7 to solve for \( x \): \[ x = \frac{\ln(9) - 4}{-7} \] So, \[ x = \frac{4 - \ln(9)}{7} \] Now you can use a calculator to find the approximate value of \( x \).
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