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Questions and Answers
When multiplying or dividing both sides of an inequality by a ______ value, the direction of the inequality remains the same.
When multiplying or dividing both sides of an inequality by a ______ value, the direction of the inequality remains the same.
positive
To solve a linear inequality, first ______ the inequality by combining like terms and removing any parentheses or fractions.
To solve a linear inequality, first ______ the inequality by combining like terms and removing any parentheses or fractions.
simplify
When solving a compound inequality, it is necessary to ______ each inequality separately.
When solving a compound inequality, it is necessary to ______ each inequality separately.
solve
When adding or subtracting the same value to both sides of an inequality, the direction of the inequality ______ the same.
When adding or subtracting the same value to both sides of an inequality, the direction of the inequality ______ the same.
After isolating the variable, it is necessary to ______ the direction of the inequality and flip it if necessary.
After isolating the variable, it is necessary to ______ the direction of the inequality and flip it if necessary.
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Study Notes
Solving Linear Inequalities
Rules for Solving Linear Inequalities
- When adding or subtracting the same value to both sides of an inequality, the direction of the inequality remains the same.
- When multiplying or dividing both sides of an inequality by a positive value, the direction of the inequality remains the same.
- When multiplying or dividing both sides of an inequality by a negative value, the direction of the inequality is reversed.
Steps for Solving Linear Inequalities
- Simplify the inequality by combining like terms and removing any parentheses or fractions.
- Isolate the variable by adding or subtracting the same value to both sides of the inequality.
- Check the direction of the inequality and flip it if necessary when multiplying or dividing by a negative value.
- Simplify the solution by writing it in its simplest form.
Examples of Solving Linear Inequalities
- Simple Inequality: 2x + 3 > 5
- Subtract 3 from both sides: 2x > 2
- Divide both sides by 2: x > 1
- Compound Inequality: -3x + 2 ≤ 5 and x - 2 > -3
- Solve each inequality separately:
- -3x + 2 ≤ 5 => -3x ≤ 3 => x ≥ -1
- x - 2 > -3 => x > -1
- Combine the solutions: x ≥ -1 and x > -1 => x > -1
- Solve each inequality separately:
- Inequality with Fractions: (x + 1)/2 ≥ 3
- Multiply both sides by 2: x + 1 ≥ 6
- Subtract 1 from both sides: x ≥ 5
Solving Linear Inequalities
Rules for Solving Linear Inequalities
- Adding or subtracting the same value to both sides of an inequality does not change the direction of the inequality.
- Multiplying or dividing both sides of an inequality by a positive value does not change the direction of the inequality.
- Multiplying or dividing both sides of an inequality by a negative value reverses the direction of the inequality.
Steps for Solving Linear Inequalities
- Simplify the inequality by combining like terms and removing any parentheses or fractions.
- Isolate the variable by adding or subtracting the same value to both sides of the inequality.
- Check the direction of the inequality and flip it if necessary when multiplying or dividing by a negative value.
- Simplify the solution by writing it in its simplest form.
Examples of Solving Linear Inequalities
- To solve a simple inequality, such as 2x + 3 > 5, subtract 3 from both sides and then divide both sides by 2 to get x > 1.
- To solve a compound inequality, such as -3x + 2 ≤ 5 and x - 2 > -3, solve each inequality separately and then combine the solutions.
- To solve an inequality with fractions, such as (x + 1)/2 ≥ 3, multiply both sides by 2 and then subtract 1 from both sides to get x ≥ 5.
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