Inequalities and linear programming : แบบฝึกหัด

Solve the inequality $$3x - 5 < x + 7$$.

Solve the quadratic inequality $$x^2 - 5x + 6 \normalsize \text{≥ } 0$$.

Consider the feasible region \(R\) defined by the inequalities:
\(x \ge 0\)
\(y \ge 0\)
\(x + y \le 15\)
\(y \le 0.5x\)
For points \((x,y)\) in \(R\), what is the maximum value of the objective function \(P = 2x + 8y\)?

Solve the compound inequality: $$2x - 1 < 5$$ and $$3x + 2 \normalsize \text{≥ } -7$$.

Solve the quadratic inequality $$(x-3)(x+2) > 0$$.

The lengths of the three sides of a triangle are \( 7 \), \( 10 \), and \( k \). Find the range of possible values of \( k \).

Solve the quadratic inequality $$x^2 - 4x - 5 < 0$$.

Solve the quadratic inequality \(3x^2 - 4x \le 4\). How many non-negative integers satisfy this inequality?

Solve the compound inequality \(5(x+1) \le 2x+14\) and \(7-2x < x-2\). Express your solution in interval notation and describe its representation on a number line.

A nutritional company produces two types of protein bars: "Power Bar" (represented by \(x\)) and "Lean Bar" (represented by \(y\)). The production is subject to the following constraints:

At least 40 Power Bars must be produced.

At least 30 Lean Bars must be produced.

The total number of protein bars produced must be at least 80.

The number of Power Bars plus twice the number of Lean Bars cannot exceed 180.

(a) Formulate a system of linear inequalities to represent the given constraints, including non-negativity constraints.

(b) On a graph paper, sketch the feasible region defined by these inequalities. Clearly label all vertices of the feasible region with their coordinates.

(c) The cost to produce a Power Bar is HK$15 and a Lean Bar is HK$20. Determine the number of each type of bar that should be produced to minimize the total production cost. State the minimum cost.