Alistair has a rectangular picture measuring 8 inches by 7 inches. he makes a frame for picture. the area of the frame is 34 square inches, and the width of the frame is x inches.

Although the question is not given,I can infere that you need to know the value of x, that is the widht of the frame.

Note that the dimensions of the rectangle covered by the frame are:

 8 + 2x and 7 + 2x, meaning that the area covered by the whole figure is:

 (8 + 2x) * (7 + 2x).

While the area of the picture is 8 in * 7 in = 56 in^2.

Then, you can state that the area of just the frame meets this equation:

area of the frame = area of the whole figure - area of the picture

34 = (8+2x)(7+2x) - 56

Now solve the equation, starting by expanding the product of the two parenthesis:

8*7 + 16x + 14x + 4x^2 - 56 = 34

4x^2 + 30x + 56 - 56 - 34 = 0

4x^2 + 30x - 34 = 0

Divide by 2 => 2x^2 + 15x - 17 = 0

Factor: (2x - 2)(x + 17/2) = 0

=> 2x - 2 = 0 => x = 1, and x + 17/2 = 0 => x = - 17/2.

Only x = 1 has a real physical meaning so, the solution is x = 1, which is the width of the frame.

the widht of the frame is 1 inch.

You can check: (8 + 2) (7 + 2) = (10)(9) = 90 = area of picture + the frame.

90 - 56 = 34 which is the area of the frame.