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Chapter 5: Problem 93

What is a system of linear inequalities?

Short Answer

Expert verified
A system of linear inequalities is a set of two or more linear inequalities having the same variables. The solution is the intersection of the solutions to each inequality, which usually forms a geometric shape.

Step by step solution

01

Understand Linear Inequality

A linear inequality is like a linear equation but instead of an equals sign, it has an inequality sign. For instance, \(2x + 3y > 6\) is a linear inequality. The solution to a linear inequality is a range of values, not just a singular point.
02

Understanding Half-Spaces

A linear inequality divides the coordinate plane into two halves, called half-spaces. For example, in the inequality \(2x + 3y > 6\), the solutions lie in the half-space where \(2x + 3y\) is more than 6. When graphed, this half-space does not include the line \(2x + 3y = 6\). If the inequality was \(2x + 3y \geq 6\), the line would be included in the half-space, and it would consist of all points that make the inequality true when substituted.
03

Understand a System of Linear Inequalities

A system of linear inequalities is a set of two or more linear inequalities with the same variables. The solution to this system is the intersection of the solutions to each inequality. This usually forms a geometric shape, graphically represented by the area where the half-spaces intersect.

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Most popular questions from this chapter

Solve the system for \(x\) and \(y\) in terms of \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2}\) and \(c_{2}\) : $$\left\\{\begin{array}{l}a_{1} x+b_{1} y=c_{1} \\ a_{2} x+b_{2} y=c_{2}\end{array}\right.$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A system of two equations in two variables whose graphs are a circle and a line can have four real ordered-pair solutions

Explain how to solve a nonlinear system using the addition method. Use \(x^{2}-y^{2}-5\) and \(3 x^{2}-2 y^{2}-19\) to illustrate your explanation.

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises \(71-72,\) you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\).

Will help you prepare for the material covered in the next section. Solve by the substitution method: $$\left\\{\begin{array}{l} 4 x+3 y=4 \\ y=2 x-7 \end{array}\right.$$
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