Solve each inequality by using algebra.
24. x 2 - 11x + 13 ≤ 25
25. -2x 2 + 3x + 4 ≥ -1
26. x 2 - 5x - 4 < -9
27. Sports A football thrown by a quarterback follows a path given by h(x) = -0.0095x 2 + x + 7, where h is the height of the ball in feet and x is the horizontal distance the ball has traveled in feet. If any height less than 10 feet can be caught or knocked down, at what distances from the quarterback can the ball be knocked down?
24. To solve the inequality x^2 - 11x + 13 ≤ 25, we start by subtracting 25 from both sides: x^2 - 11x + 13 - 25 ≤ 0. Simplifying, we get x^2 - 11x - 12 ≤ 0.
Next, we factor the quadratic equation (x - 12)(x + 1) ≤ 0. To determine the intervals where the inequality holds true, we look at the signs of the factors.
Setting (x - 12) ≤ 0, we find x ≤ 12.
Setting (x + 1) ≤ 0, we find x ≤ -1.
So the solution to the inequality is x ≤ -1 or x ≤ 12.
25. For the inequality -2x^2 + 3x + 4 ≥ -1, the first step is to simplify by adding 1 to both sides: -2x^2 + 3x + 4 + 1 ≥ 0. This becomes -2x^2 + 3x + 5 ≥ 0.
Next, we factor the quadratic equation or solve it using the quadratic formula. However, in this case, the quadratic cannot be factored easily. So we solve it using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
Using the formula, we find x = (-3 ± √(3^2 - 4(-2)(5))) / (2(-2)), which simplifies to x = (-3 ± √(9 + 40)) / (-4).
Simplifying further, we get x = (-3 ± √49) / (-4). This gives us two answers: x = (-3 + 7) / (-4) = 1 and x = (-3 - 7) / (-4) = 2.
So the solution to the inequality is x ≥ 1 or x ≤ 2.
26. To solve the inequality x^2 - 5x - 4 < -9, the first step is to add 9 to both sides: x^2 - 5x - 4 + 9 < 0. Simplifying, we get x^2 - 5x + 5 < 0.
Next, we factor the quadratic equation (x - 1)(x - 5) < 0. To determine the intervals where the inequality holds true, we look at the signs of the factors.
Setting (x - 1) < 0, we find x < 1.
Setting (x - 5) < 0, we find x < 5.
So the solution to the inequality is x < 1 and x < 5.
27. In this problem, we need to find the distances from the quarterback at which the ball can be caught or knocked down, given the height of the ball.
The inequality given is h(x) = -0.0095x^2 + x + 7 ≤ 10. We are looking for the values of x that satisfy this inequality.
To solve this inequality, we subtract 10 from both sides: -0.0095x^2 + x + 7 - 10 ≤ 0. Simplifying, we get -0.0095x^2 + x - 3 ≤ 0.
Next, we can either try to factor the quadratic equation or solve it using the quadratic formula. However, in this case, the quadratic cannot be factored easily. So we solve it using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
Using the formula, we find x = (-1 ± √(1 - 4(-0.0095)(-3))) / (2(-0.0095)).
Simplifying further, we get x = (-1 ± √(1 + 0.114)) / (-0.019). This gives us two answers: x ≈ (-1 + √1.114) / (-0.019) and x ≈ (-1 - √1.114) / (-0.019).
So the distances from the quarterback at which the ball can be caught or knocked down are approximately x ≈ (-1 + √1.114) / (-0.019) feet and x ≈ (-1 - √1.114) / (-0.019) feet.