1.02g Inequalities: linear and quadratic in single variable

Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 \(\square\) Row 2 \(\square\square\) Row 3 \(\square\square\square\) She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.

1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. [3]

Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.

1. Find the total number of sticks Ann uses in making these 10 rows. [3]

Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the \((k + 1)\)th row,

1. show that \(k\) satisfies \((3k - 100)(k + 35) < 0\). [4]
2. Find the value of \(k\). [2]

The equation \(x^2 + 5kx + 2k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k(25k - 8) \geq 0\). [2]
2. Hence find the set of possible values of \(k\). [4]
3. Write down the values of \(k\) for which the equation \(x^2 + 5kx + 2k = 0\) has equal roots. [1]

Find the set of values for \(x\) for which

1. \(6x - 7 < 2x + 3\), [2]
2. \(2x^2 - 11x + 5 < 0\), [4]
3. both \(6x - 7 < 2x + 3\) and \(2x^2 - 11x + 5 < 0\). [1]

The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m,

1. form a linear inequality in \(x\). [2]

Given that the area of the pitch must be greater than 4800 m²,

1. form a quadratic inequality in \(x\). [2]
2. by solving your inequalities, find the set of possible values of \(x\). [4]

Find the set of values of \(x\) for which $$\frac{3}{x+3} > \frac{x-4}{x}.$$ [7]

Use algebra to find the set of values of \(x\) for which $$\frac{6x}{3 - x} > \frac{x + 1}{1}$$ [7]

Find the set of values for which $$|x - 1| > 6x - 1.$$ [5]

Using algebra, find the set of values of \(x\) for which $$2x - 5 > \frac{3}{x}.$$ [7]

Solve the inequality \(\frac{1}{2x + 1} > \frac{x}{3x - 2}\). [6]

Find the complete set of values of \(x\) for which $$|x^2 - 2| > 2x.$$ [7]

The equation \(2x^2 + 2kx + (k + 2) = 0\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies $$k^2 - 2k - 4 > 0$$ [3]
2. Find the set of possible values of \(k\). [4]

Solve the inequalities

1. \(1 < 4x - 9 < 5\), [3]
2. \(y^2 \geq 4y + 5\). [5]

Solve the inequalities

1. \(3 - 8x > 4\), [2]
2. \((2x - 4)(x - 3) \leq 12\). [5]