We are just starting this section, and I'm confused...Please help me understand how to solve this inequality:

(one fifth r is less than two)

1/5r < 2

Also, simplify: (the #s are exponents)

z3 . z3

x12/x3

By 1/5r you mean r/5 or 1/(5r)?

z^3 . z^3 = z^6

x^12/x^3 = x^9

If you mean r/5<2, multiply each side by 5, leaving you with r<10. If you mean 1/(5r)<2, multiply each side by 5r, leaving you with 1<10r. Then divide each side by 10 and r>1/10.

To solve the inequality (1/5r < 2), we need to isolate the variable "r" on one side of the inequality symbol.

If by 1/5r you mean r/5, then the inequality can be rewritten as:

r/5 < 2

To solve for "r", we can multiply both sides of the inequality by 5, which gives us:

5 * (r/5) < 2 * 5

This simplifies to:

r < 10

So, if 1/5r means r/5, the solution is r < 10.

On the other hand, if by 1/5r you mean 1/(5r), then the inequality becomes:

1/(5r) < 2

To eliminate the fraction, we can multiply both sides by 5r:

5r*(1/(5r)) < 2*(5r)

This simplifies to:
1 < 10r

To isolate "r", we can divide both sides of the inequality by 10, giving us:

1/10 < r

So, if 1/5r means 1/(5r), the solution is 1/10 < r.

For the simplification questions:

z^3 . z^3 can be simplified by adding the exponents since we are multiplying the same base:

z^(3+3) = z^6

Thus, z^3 . z^3 simplifies to z^6.

For x^12/x^3, we can simplify by subtracting the exponents since we are dividing the same base:

x^(12-3) = x^9

Thus, x^12/x^3 simplifies to x^9.