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Chapter 1: Problem 114

Describe ways in which solving a linear inequality is different than solving a linear equation.

Short Answer

Expert verified
The main differences are: (1) The solution to a linear equation is a specific value, while the solution to a linear inequality is a range of values; (2) Multiplying or dividing by a negative number in an inequality causes the inequality sign to change direction, but such a rule doesn't apply to equations.

Step by step solution

01

Understanding Linear Equality

The first task is to understand what linear equalities are. A linear equality is an equation of the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are real numbers and \(A\) and \(B\) are not all zero. The objective here is to find a value for \(x\) that will make the equation true. This is usually accomplished by isolating \(x\) on one side of the equation.
02

Understanding Linear Inequality

After understanding linear equalities, the focus moves to linear inequalities. A linear inequality is similar to a linear equality, but instead of equal to (\(=\)), it uses inequality signs (\(>, <, \geq, \leq\)). Hence its form is \(Ax + By > C\) or \(Ax + By < C\) or \(Ax + By \geq C\) or \(Ax + By \leq C\). The objective here is to find a range of values for \(x\) that will make the inequality true. This is usually accomplished by isolating \(x\) on one side of the inequality.
03

Highlighting the Differences

The main differences between a linear equation and a linear inequality lie in the solutions and solving processes. When solving a linear equation, the goal is to find a specific value that makes the equation true; while in a linear inequality, the goal is to find a range of values that make the inequality true. Also, when you multiply or divide by a negative number while solving an inequality, the sign of the inequality changes, something that doesn't apply for equations.

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Most popular questions from this chapter

Writing in Mathematics What is a quadratic inequality?

Describe similarities and differences between the solutions of $$ (x-2)(x+5) \geq 0 \text { and } \frac{x-2}{x+5} \geq 0 $$

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