Question

(2/3)^x+8<=(3/2)^2-7x

Answer 1

Надеюсь это были дроби

Answer 2

To solve the inequality (2/3)^x + 8 <= (3/2)^2 - 7x, we need to find the values of x that satisfy this inequality. Let's work through the steps to solve it:

Step 1: Simplify both sides of the inequality. First, simplify (3/2)^2:

(3/2)^2 = (3/2)*(3/2) = 9/4

Now the inequality becomes:

(2/3)^x + 8 <= 9/4 - 7x

Step 2: Get rid of the constant on the left side (8) by subtracting 8 from both sides:

(2/3)^x <= 9/4 - 7x - 8

Step 3: Combine the constant terms on the right side:

(2/3)^x <= 9/4 - 7x - 32/4

(2/3)^x <= (9 - 32 - 28x) / 4

(2/3)^x <= (-23 - 28x) / 4

Step 4: Now, we need to get rid of the fraction on the left side, so take the logarithm (base 2/3) of both sides to get rid of the exponent:

log base (2/3) [(2/3)^x] <= log base (2/3) [(-23 - 28x) / 4]

Since log base (a) [a^x] = x, we have:

x <= log base (2/3) [(-23 - 28x) / 4]

Step 5: Solve for x.

To solve for x, we'll isolate x on one side. However, since there is no simple algebraic way to do this, we can use numerical methods or graphing software to approximate the solutions. It's important to remember that exponential equations can have multiple solutions, so there might be more than one valid value of x that satisfies the inequality.

For example, you can use graphing software or calculators to plot the two functions:

y1 = (2/3)^x y2 = (-23 - 28x) / 4

The values of x where y1 is less than or equal to y2 will be the solutions to the inequality.

Keep in mind that approximations are usually sufficient unless you need exact values. If you have access to graphing software, you can graph the two functions and find the regions where the inequality is satisfied. Otherwise, you can use numerical methods like the bisection method or Newton-Raphson method to approximate the solutions.