Math Problem Statement
Solve each non-linear inequality. Write the final answer in interval notation.
x2 – 2x > 35 2x2 + 7x – 4 ≥ 0 2x2 – 5x < – 3
Solution
Let's solve each inequality step-by-step and write the solutions in interval notation.
1. Solve:
Step 1: Move all terms to one side of the inequality.
Step 2: Factor the quadratic expression.
We need to find two numbers that multiply to and add to . The numbers are and .
Step 3: Determine the critical points.
The critical points are and .
Step 4: Analyze the sign of the expression around the critical points.
We now check the sign of in the intervals , , and .
- For , pick :
- For , pick :
- For , pick :
Step 5: Write the solution.
The expression is positive when .
Final answer:
2. Solve:
Step 1: Factor the quadratic expression.
We factor .
Step 2: Determine the critical points.
Set each factor equal to zero: Thus, the critical points are and .
Step 3: Analyze the sign of the expression around the critical points.
We check the sign of in the intervals , , and .
- For , pick :
- For , pick :
- For , pick :
Step 4: Write the solution, including the points where the expression equals zero.
The expression is non-negative when .
Final answer:
3. Solve:
Step 1: Move all terms to one side of the inequality.
Step 2: Factor the quadratic expression.
We need to factor .
Step 3: Determine the critical points.
Set each factor equal to zero: Thus, the critical points are and .
Step 4: Analyze the sign of the expression around the critical points.
We check the sign of in the intervals , , and .
- For , pick :
- For , pick : ((2(1.25) - 3)(1.25 - 1
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Inequalities
Factoring
Critical Points
Interval Notation
Formulas
Quadratic equation: ax^2 + bx + c = 0
Factoring of quadratic expressions
Theorems
Sign Analysis for Quadratic Inequalities
Zero Product Property
Suitable Grade Level
Grades 9-12