1.The point P divides the interval joining A(-1,-2) to B(9,3) internally in the ratio 4:1. Find the coordinates of P.

2. Solve.
4-x/x<1

My answer: x>2

P is 1/5 of the way from A to B.

So, add 1/5 of the difference to to get

P = (-1 + 1/5 (9+1) , -2 + 1/5 (3+2)) = (1,-1)

Or, if P is 4/5 of the way, P=(7,2)

To solve the first question, we need to find the coordinates of the point P that divides the interval AB internally in the ratio 4:1.

Step 1: Find the difference in x-coordinates and y-coordinates between A and B.
Δx = x_b - x_a = 9 - (-1) = 10
Δy = y_b - y_a = 3 - (-2) = 5

Step 2: Multiply the difference in x-coordinates and y-coordinates by the ratios (4:1).
4 * Δx = 4 * 10 = 40
4 * Δy = 4 * 5 = 20

Step 3: Add the products to the coordinates of A to get the coordinates of P.
x_p = x_a + 40 = -1 + 40 = 39
y_p = y_a + 20 = -2 + 20 = 18

Therefore, the coordinates of point P are (39, 18).

Moving on to the second question, let's solve the inequality.

Step 1: Simplify the inequality expression.
4 - x / x < 1

Step 2: Get rid of the denominator by multiplying both sides of the inequality by x.
x * (4 - x) / x < 1 * x

Simplifying further, we have:
4 - x < x

Step 3: Simplify and isolate the variable on one side.
-x - x < -4
-2x < -4

Step 4: Divide both sides of the inequality by -2. Note that when dividing by a negative number, the inequality sign flips direction.
x > -4 / -2
x > 2

Therefore, the solution to the inequality is x > 2.