Podcast
Questions and Answers
What is the first step in solving the inequality $3x + 8 > 2x + 2$?
What is the first step in solving the inequality $3x + 8 > 2x + 2$?
Solve the inequality $0.5x + 0.2 > 0.3$.
Solve the inequality $0.5x + 0.2 > 0.3$.
To solve the inequality $2x/3 - 3/4 > 0$, we need to multiply both sides by:
To solve the inequality $2x/3 - 3/4 > 0$, we need to multiply both sides by:
Solve the inequality $3x + 2 > 2x - 4$.
Solve the inequality $3x + 2 > 2x - 4$.
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The inequality $|2x - 3| > 3$ is an example of:
The inequality $|2x - 3| > 3$ is an example of:
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What does the inequality 3x^2 + 2x > 2x^2 + 2 simplify to?
What does the inequality 3x^2 + 2x > 2x^2 + 2 simplify to?
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In the inequality √(x + 3) > 3, what step is taken next after squaring both sides?
In the inequality √(x + 3) > 3, what step is taken next after squaring both sides?
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What is the correct way to solve the inequality log10(x + 3) > 2?
What is the correct way to solve the inequality log10(x + 3) > 2?
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Why is the inequality sin⁻¹(x) > 0 only true for x greater than 0?
Why is the inequality sin⁻¹(x) > 0 only true for x greater than 0?
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Which of the following statements about absolute value is true?
Which of the following statements about absolute value is true?
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If $|x| < 3$, which of the following inequalities is true?
If $|x| < 3$, which of the following inequalities is true?
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What is the absolute value of the complex number $3 + 4i$?
What is the absolute value of the complex number $3 + 4i$?
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In Python, how would you find the absolute value of $-7.2$?
In Python, how would you find the absolute value of $-7.2$?
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Which of the following inequalities is equivalent to $|x - 2| \leq 3$?
Which of the following inequalities is equivalent to $|x - 2| \leq 3$?
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Study Notes
Solving Inequalities
Solving inequalities is a process of finding all the possible solutions to an inequality. An inequality is a mathematical equation where different symbols such as >, <, ≥ or ≤ separate two quantities. To solve an inequality, you want either side of the inequality to remain unchanged for all possible values of x. This means that if the solution involves multiplying or dividing both sides by a negative number, we need to reverse the direction symbol. Here's how you can solve simple and complex inequalities:
Simple Inequality
A problem like this: 3x + 8 > 2x + 2 is a simple inequality. It's possible to solve this problem by subtracting 3x from both sides of the equation, and then subtracting 8 from both sides. The result is 3x - 3x + 8 > 2x - 3x + 2, which simplifies to 8 > 2.
Complex Inequality
A problem like this: 3x + 8 > 2x + 2 is a simple inequality. It's possible to solve this problem by subtracting 3x from both sides of the equation, and then subtracting 8 from both sides. The result is 3x - 3x + 8 > 2x - 3x + 2, which simplifies to 8 > 2.
Inequality Involving Decimals
A problem like this: 0.5x + 0.2 > 0.3 is a simple inequality. To solve this problem, we subtract 0.5x from both sides of the equation, and then subtract 0.2 from both sides. The result is 0.5x - 0.5x + 0.2 > 0.3 - 0.2, which simplifies to 0.2 > 0.1.
Inequality Involving Fractions
A problem like this: 2x/3 - 3/4 > 0 is a simple inequality. To solve this problem, we multiply both sides of the equation by 12, which is the least common multiple of 3 and 4. The result is 8x - 9 > 0.
Inequality Involving Variables on Both Sides
A problem like this: 3x + 2 > 2x - 4 is a complex inequality. To solve this problem, we subtract 2x from both sides of the equation, and then subtract 2 from both sides. The result is 3x - 2x + 2 > 2x - 2x - 4, which simplifies to 5 > -4.
Inequality Involving Absolute Values
A problem like this: |2x - 3| > 3 is a complex inequality. To solve this problem, we need to consider two cases: when 2x - 3 is negative and when 2x - 3 is positive. When 2x - 3 is negative, the absolute value is -(2x - 3). When 2x - 3 is positive, the absolute value is 2x - 3. In either case, the inequality becomes -(2x - 3) > -(3), which simplifies to 2x - 3 > 3.
Inequality Involving Exponents
A problem like this: 3x^2 + 2x > 2x^2 + 2 is a complex inequality. To solve this problem, we subtract 2x^2 from both sides of the equation, and then subtract 2 from both sides. The result is 3x^2 - 2x^2 + 2 > 2x^2 - 2x^2 + 2, which simplifies to 3 > 0.
Inequality Involving Square Roots
A problem like this: √(x + 3) > 3 is a complex inequality. To solve this problem, we need to square both sides of the equation. The result is (√(x + 3))^2 > 3^2, which simplifies to x + 3 > 9.
Inequality Involving Logarithms
A problem like this: log10(x + 3) > 2 is a complex inequality. To solve this problem, we need to raise both sides of the equation to the power of 10. The result is 10^(log10(x + 3)) > 10^2, which simplifies to x + 3 > 100.
Inequality Involving Trigonometry
A problem like this: sin(2x) > 0 is a complex inequality. To solve this problem, we need to note that the sine function is always positive in the first and second quadrants. Therefore, the inequality always holds true.
Inequality Involving Hyperbolic Functions
A problem like this: cosh(x) > 0 is a complex inequality. To solve this problem, we need to note that the hyperbolic cosine function is always positive. Therefore, the inequality always holds true.
Inequality Involving Inverse Trigonometric Functions
A problem like this: sin⁻¹(x) > 0 is a complex inequality. To solve this problem, we need to note that the inverse sine function is always positive in the first quadrant. Therefore, the inequality holds true for x > 0.
Inequality Involving Exponential Functions with Different Base
A problem like this: 2^x < 3^x is a complex inequality. To solve this problem, we can take natural logarithms of both sides of the equation. The result is ln(2^x) < ln(3^x), which simplifies to xln(2) < xln(3), which further simplifies to ln(2) < ln(3). This inequality is actually false, so there are no solutions to this problem.
In conclusion, solving inequalities involves manipulating algebraic expressions while ensuring that the direction symbol remains unchanged for all possible values of x. It requires careful consideration of different cases and principles of algebra such as properties of equality, exponents, and functions.
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Description
This quiz covers the process of solving inequalities, ranging from simple cases involving basic arithmetic operations to complex scenarios involving absolute values, exponents, square roots, logarithms, trigonometry, and hyperbolic functions. Learn how to determine all the possible solutions to various types of inequalities by manipulating algebraic expressions and understanding key algebra principles.
