Linear Inequalities and Systems of Linear Inequalities

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What is a linear inequality?

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A linear inequality is a mathematical statement that compares a linear expression to a value using inequality symbols such as <, >, ≤, or ≥.

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How do you solve a linear inequality?

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To solve a linear inequality, isolate the variable on one side of the inequality using inverse operations, similar to solving an equation, while remembering to reverse the inequality sign if you multiply or divide by a negative number.

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What does it mean for a point to be a solution to a system of linear inequalities?

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A point is a solution to a system of linear inequalities if it satisfies all inequalities in the system, meaning it lies in the region defined by the inequalities.

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What is the significance of a solid line in graphing linear inequalities?

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A solid line indicates that the points on the line are included in the solution set, which occurs when the inequality symbol is ≤ or ≥.

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What is the significance of a dashed line in graphing linear inequalities?

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A dashed line indicates that the points on the line are not included in the solution set, which occurs when the inequality symbol is < or >.

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How can you determine if a point is a solution to a system of linear inequalities?

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To determine if a point is a solution, substitute the coordinates of the point into each inequality in the system. If the point satisfies all inequalities, it is a solution.

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What is the graphical representation of a linear inequality?

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The graphical representation of a linear inequality is a shaded region on one side of a line, which shows all the possible solutions that satisfy the inequality.

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What does the term 'boundary line' refer to in the context of linear inequalities?

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The boundary line is the line that represents the equation of the linear inequality, dividing the coordinate plane into two regions.

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How do you write an inequality from a word problem?

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To write an inequality from a word problem, identify the variable, determine the relationship described (greater than, less than, etc.), and express the situation mathematically.

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What is the difference between a system of linear equations and a system of linear inequalities?

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A system of linear equations consists of equations that are equal, while a system of linear inequalities consists of inequalities that define a range of possible solutions.