Investigating Hard Inequalities: Understanding Part (f)

I am still confused as to how the markscheme answers have come about which I attached above.In summary, the task is to find the smallest value of n such that x_n is less than n+0.05. This problem involves solving a quadratic equation, 2x_n^2- (2n-1)x- (n+1)= 0, which has been proven in part (e). The solution to this equation can be compared to n+0.05 to determine the smallest value of n. The markscheme provided shows the step-by-step process for solving the equation and finding the smallest value of n.
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  • #2
If (f) is the part you are having trouble with, then presumably you have already proved that [itex]2x_n^2- (2n-1)x- (n+1)= 0[/itex] (part (e)). Now you want to find the smallest n such that [itex]x_n< n+ 0.05[/itex]. You could, for example, solve that using the quadrative formula and compare the solutions to n+ 0.05. Have you calculated some values of [itex]x_n[/itex]? What are [itex]x_0[/itex] [itex]x_1[/itex], etc.?
 
  • #3
Thanks for the help. I am still confused as to how the markscheme answers have come about which I attached above.

Thanks
 

1. How do you solve a hard inequality question?

To solve a hard inequality question, you need to first identify the type of inequality (e.g. linear, quadratic, absolute value), then use algebraic techniques such as factoring, completing the square, or using the quadratic formula to isolate the variable on one side of the inequality.

2. What are some common mistakes to avoid when solving hard inequalities?

Some common mistakes to avoid when solving hard inequalities include forgetting to flip the inequality sign when multiplying or dividing by a negative number, not properly distributing negative signs, and not considering restrictions on the variable (e.g. when dealing with absolute value inequalities).

3. How do you know when to use a test value in solving hard inequalities?

A test value is used to determine the direction of the inequality sign in cases where the variable is between two values (e.g. 0 < x < 5). You can choose any value within the given range, plug it into the inequality, and see if it satisfies the inequality. If it does, then that direction of the inequality sign is correct.

4. Can you graph hard inequalities?

Yes, hard inequalities can be graphed on a number line or on a coordinate plane. The solution set of the inequality will be represented by a shaded region or a series of points on the graph.

5. How can hard inequality questions be applied in real-life situations?

Hard inequalities can be used to model and solve real-life problems such as determining the maximum or minimum value of a quantity, finding the range of possible values for a variable, or determining the conditions for a certain event to occur.

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