GED · Mathematical Reasoning · 6 min read
Linear Inequalities and Systems of Equations
- Understand inequality symbols, solve linear inequalities using inverse operations, and graph solution sets accurately on a number line.
- Identify and classify solutions to systems of linear equations graphically and algebraically as having one solution, no solution, or infinitely many solutions.
- Solve systems of linear equations using substitution, elimination, and coordinate testing, and set up equations for real-world word problems.
Understanding and Solving Linear Inequalities
An inequality compares two mathematical expressions using the symbols <, >, ≤, or ≥. When working with these mathematical statements, the symbol < means 'is strictly less than', and > means 'is strictly greater than'. The symbol ≤ means 'is less than or equal to', and ≥ means 'is greater than or equal to'. Solving a linear inequality means finding all values of the variable that make the inequality true.
To solve a linear inequality, simplify each side and use inverse operations to isolate the variable. Terms with the same variable on the same side of an inequality should be combined before moving terms across the symbol. If an inequality contains parentheses, you can often use the distributive property to simplify it before isolating the variable. When a variable appears on both sides of an inequality, subtract or add one variable term so it appears on only one side.
As you isolate your variable, remember the rules for working with both sides. Adding the same number to both sides of an inequality keeps the inequality true and does not flip the sign. Similarly, subtracting the same number from both sides of an inequality does not flip the inequality sign. Multiplying or dividing both sides of an inequality by a positive number does not change the direction of the sign. However, multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. When reversing an inequality symbol, < becomes >, and ≤ becomes ≥.
Always take time to format your final answer clearly. Rewriting a solution with the variable on the left, such as x > 5 instead of 5 < x, makes it much easier to interpret and graph. Keeping the variable on the left side helps you read the solution set naturally and avoid simple direction mistakes.
Common mistakes
- Forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number.
- Using an open circle for inclusive inequalities (≤, ≥) or a solid circle for strict inequalities (<, >) on a number line.
- Checking a proposed coordinate solution in only one equation instead of plugging it into both equations to confirm it satisfies both.
- Continuing to solve for the second variable when the problem asks only for the value of the first variable.
Key points
- An inequality compares expressions using <, >, ≤, or ≥, and solving it means finding all values that make the statement true.
- Isolate variables using inverse operations, reversing the inequality symbol (< becomes >, ≤ becomes ≥) whenever you multiply or divide by a negative number.
- On a number line, strict inequalities (<, >) use open circles, inclusive inequalities (≤, ≥) use solid circles, and variable-on-the-left statements point right for > or ≥ and left for < or ≤.
- Systems of two linear equations can have one solution (intersecting lines), no solution (parallel lines producing false statements like 0 = 8), or infinitely many solutions (coincident lines producing identities like 0 = 0).
- Both substitution and elimination yield identical solutions; substitution is ideal when a variable has a coefficient of 1 or -1, while elimination works best in standard form Ax + By = C using opposite coefficients like 4y and -4y.
- Word problems can model item quantities and monetary values, mixture amounts, break-even points where total revenue equals total cost, and motion with or against a current using distance = rate × time.
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