Solve 0.6^x < 0.8
So here's what I did:
x log0.6 < log 0.8
x < log 0.8/ log 0.6
x < 0.437
But that's not the correct answer, it's:
x > 0.437
Can someone explain why it changes < from to > ? Thanks in advance!
let's pick it up from
x log0.6 < log 0.8 , now put in the actual values
x(-.22185) < -.09691
to get x, we would have to divide both sides by -.22185
BUT, remember if you divide an inequality by a negative
you must reverse the inequality sign
x(-.22185)/-.22185 > -.09691/-.22185
x > .4368
Consider the graph of y = .6^x
see here:
www.wolframalpha.com/input/?i=plot+y+%3D+(.6)%5Ex+,+y+%3D+.8
notice that the curve is below y = .8 for all x's > .4368
To solve the inequality 0.6^x < 0.8, you correctly took the logarithm of both sides of the inequality. However, there seems to be a small error in your calculations, which led to the incorrect inequality sign.
Let's break down the steps to solve the inequality correctly:
1. Start with the inequality: 0.6^x < 0.8
2. Take the logarithm of both sides of the inequality.
We can choose any logarithmic base, but let's use the natural logarithm (ln) for simplicity:
ln(0.6^x) < ln(0.8)
3. Apply the logarithmic property, which states that the logarithm of a power is equal to the product of the exponent multiplied by the logarithm of the base:
x * ln(0.6) < ln(0.8)
4. Divide both sides of the inequality by ln(0.6) to isolate x:
x < ln(0.8) / ln(0.6)
So far, you did everything right. However, the value you obtained for x (0.437) is not accurate.
Calculating the exact value, we have:
x < ln(0.8) / ln(0.6)
x < -0.22314 / -0.51082
x < 0.43691
Rounding to the nearest thousandth, x is approximately 0.437.
Therefore, the correct inequality is:
x < 0.437
This means that any value of x less than 0.437 will satisfy the original inequality.
Sometimes, when we divide or multiply both sides of an inequality by a negative number, the inequality sign flips. However, in this case, since we divided both sides by positive values (ln(0.6) and ln(0.8)), the inequality sign remains the same as the original inequality.
Hope this explanation clarifies the correct result for you!