Inequalities: แบบฝึกหัด

What is the solution to the inequality \( 4 - x > 2(x - 1) \)?

A rectangular garden has a length of \( (x + 5) \) m and a width of \( (x - 2) \) m. If the area of the garden must be at least \( 30 \text{ m}^2 \), find the range of possible values for \( x \).

Solve the inequality \( (x + 4)(x - 1) \ge 0 \).

Find the values of \( x \) that satisfy the simultaneous inequalities:
\( 2x + 1 > 5 \)
\( x^2 - 10x + 21 < 0 \)

Find the set of values of \( k \) for which the quadratic equation \( x^2 + kx + 9 = 0 \) has real distinct roots.

Find the set of values of \(x\) for which \(x^{2} - 7x + 10 > 0\).

Solve the inequality \(3(x - 4) < 15\) and express your answer in set notation.

Find the set of values of \( k \) for which the quadratic equation \( 3x^2 - 6x + k = 0 \) has no real roots. Express your answer using interval notation.

(a) Solve the linear inequality:
\( 5x - 3 < 2(x + 6) \)

(b) Solve the quadratic inequality:
\( x^2 - 2x - 15 \geq 0 \)

(c) Find the set of values of \( x \) that satisfy both the inequality in part (a) and the inequality in part (b). Express your answer using set notation.