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Solving Linear and Quadratic Inequalities: Exercises for Formal Evaluation
Published on April 12, 2026
Algebra
This problem set includes a variety of linear and quadratic inequalities, ranging from simple to complex. The solution approach involves algebraic manipulation and understanding the properties of quadratic functions.
Question
Answer
The user wants me to solve the math problems presented in the image. The image contains a list of inequalities. I will solve each inequality individually.
Let’s solve each inequality step by step:
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Subtractfrom both sides:
Subtract 5 from both sides: -
Distribute:
Combine like terms:
Addto both sides:
Subtract 6 from both sides:
Divide by 2:or -
Divide by 9: -
Distribute:
Addto both sides:
Add 2 to both sides:
Divide by 3: -
Distribute:
Subtractfrom both sides:
Add 5 to both sides:
Divide by 3: -
Distribute:
Combine like terms:
Subtractfrom both sides:
Subtract 2 from both sides:or -
Distribute:
Combine like terms:
Subtract 6 from both sides:
Multiply by -1 and reverse the inequality sign: -
Distribute:
Combine like terms:
Addto both sides:
Subtract 5 from both sides:
Divide by 8: -
Distribute:
Combine like terms:
Subtractfrom both sides:
Subtract 66 from both sides:
Divide by 2:or -
Distribute:
Combine like terms:
Subtractfrom both sides:
This statement is always true, so the solution is all real numbers. -
This is a quadratic inequality. We first find the roots ofusing the quadratic formula .
Here,, , .
The roots areand .
Since the parabola opens upwards (coefficient ofis positive), the inequality is satisfied when or . -
Factor the difference of squares:
The roots areand .
The inequality is satisfied whenor . -
Factor the quadratic:
The roots areand .
The inequality is satisfied when. -
Sinceis always non-negative, is always greater than or equal to 16.
Therefore,has no real solutions. -
This is an equation, not an inequality. Let’s assume it’sor .
Let’s check the discriminant of: .
Since the discriminant is negative and the coefficient ofis positive, the quadratic is always positive for all real values of .
So, if the problem was, there would be no solution.
If the problem was, the solution would be all real numbers.
If it is an equation, then there are no real solutions. Assuming it’s an inequality for the context of the other problems. Given the other problems are inequalities and it has an equals sign here, and no inequality symbol, it’s likely a typo for an inequality. Let’s assume for now that it’s an equation, and thus has no real solutions. -
Factor the quadratic:
The roots areand .
The inequality is satisfied when. -
Let’s check the discriminant of: .
Since the discriminant is negative and the coefficient ofis positive, the quadratic is always positive for all real values of .
Therefore,is true for all real numbers. -
The roots areand .
The inequality is satisfied whenor . -
Factor the quadratic:
The roots areand .
The inequality is satisfied whenor .
Summary of Solutions:
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- All real numbers
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or -
or -
- No solution
- Assuming it’s an equation: No real solution. If it’s an inequality, more information is needed or a clarification of the intended inequality symbol. Based on context, let’s consider it as an equation with no real solution.
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- All real numbers
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or -
or
The final answer is $\boxed{
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- All real numbers
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or -
or -
- No solution
- No real solution (assuming equation)
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- All real numbers
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or -
or
}$.