Inequalities

Question 1

How do you solve linear inequalities analytically?

Answer 1

1. Isolate the Variable: Move all terms involving the variable to one side of the inequality.
2. Simplify: Perform basic operations (addition, subtraction, multiplication, or division) to solve for the variable.
3. Check Direction: If you multiply or divide by a negative number, reverse the inequality sign.
4. Write the Solution: Express the solution in interval notation or as an inequality.

Question 2

How do you solve quadratic inequalities with integral roots?

Answer 2

1. Rewrite as an Equation: Set the quadratic expression equal to zero and solve for the roots.
2. Determine Intervals: Use the roots to divide the number line into intervals.
3. Test Intervals: Substitute values from each interval into the original inequality to check which intervals satisfy the inequality.
4. Write the Solution: Express the solution using inequality notation or interval notation.

Question 2

How do you solve linear inequalities graphically?

Answer 2

1. Rewrite the Inequality: Express the inequality in the form 𝑦 ≤ (or≥, <,>) 𝑚𝑥 +𝑐.
2. Graph the Line: Plot the line corresponding to the equation 𝑦 = 𝑚𝑥 + 𝑐.
3. Shade the Region: Depending on the inequality sign, shade the region above or below the line.
4. Interpret: The shaded region represents the solution set.

Question 3

How do you interpret the graph of a quadratic inequality?

Answer 3

1. Graph the Parabola: Sketch the graph of the quadratic equation.
2. Determine Regions: Identify the regions where the parabola is above or below the x-axis.
3. Interpret: Depending on the inequality sign, select the appropriate region as the solution set.