Analyzing the Given Phrases

Oct 13, 2025 by windailys.com 26 views

In mathematics, inequalities are statements that compare two values or expressions, indicating whether one is greater than, less than, or equal to the other. In this article, we will explore the concept of inequalities and identify which of the given phrases represent inequalities.

What are Inequalities?

Inequalities are mathematical statements that compare two values or expressions using symbols such as <, >, ≤, or ≥. These symbols indicate the relationship between the two values or expressions. For example:

${x < y}$ means that ${x}$ is less than ${y}$
${x > y}$ means that ${x}$ is greater than ${y}$
${x ≤ y}$ means that ${x}$ is less than or equal to ${y}$
${x ≥ y}$ means that ${x}$ is greater than or equal to ${y}$

Examples of Inequalities

Here are some examples of inequalities:

${2x + 3 > 5}$
${x - 2 < 3}$
${4x ≥ 12}$
${x + 1 ≤ 4}$

Analyzing the Given Phrases

Now, let's analyze the given phrases and identify which ones represent inequalities.

Choice A: ${5=\dfrac{55}{11}}$

This phrase represents an equation, not an inequality. An equation is a statement that says two values or expressions are equal. In this case, the fraction ${\dfrac{55}{11}}$ is equal to 5.

Choice B: ${-2<2\text{}}$

This phrase represents an inequality. The symbol < indicates that -2 is less than 2.

Choice C: ${10-6q}$

This phrase does not represent an inequality. It is an expression with a variable ${q}$ , but it does not compare two values or expressions.

Choice D: ${7w>5.6\cdot

This phrase represents an inequality. The symbol > indicates that 7w is greater than 5.6.

Conclusion

In conclusion, the phrases that represent inequalities are:

Choice B: [-2<2\text{}}$
Choice D: [7w>5.6\cdot

These phrases use the symbols < and > to compare two values or expressions, indicating that one is less than or greater than the other.

Discussion

Inequalities are an essential concept in mathematics, and they have numerous applications in real-life situations. Inequalities can be used to model real-world problems, such as comparing the cost of two products or determining the maximum or minimum value of a function.

Common Types of Inequalities

There are several types of inequalities, including:

Linear inequalities: These are inequalities that involve a linear expression, such as ${x + 2 > 3}$ .
Quadratic inequalities: These are inequalities that involve a quadratic expression, such as ${x^2 + 4x + 4 > 0}$ .
Polynomial inequalities: These are inequalities that involve a polynomial expression, such as ${x^3 + 2x^2 - 3x - 1 > 0}$ .

Solving Inequalities

Solving inequalities involves finding the values of the variable that satisfy the inequality. There are several methods for solving inequalities, including:

Graphing: This involves graphing the inequality on a number line or coordinate plane.
Factoring: This involves factoring the inequality to find the values of the variable that satisfy the inequality.
Using algebraic methods: This involves using algebraic methods, such as adding or subtracting the same value to both sides of the inequality.

Real-World Applications of Inequalities

Inequalities have numerous real-world applications, including:

Comparing costs: Inequalities can be used to compare the cost of two products or services.
Determining maximum or minimum values: Inequalities can be used to determine the maximum or minimum value of a function.
Modeling real-world problems: Inequalities can be used to model real-world problems, such as determining the maximum or minimum value of a function.

Conclusion

In this article, we will answer some of the most frequently asked questions about inequalities.

Q: What is the difference between an equation and an inequality?

A: An equation is a statement that says two values or expressions are equal, while an inequality is a statement that compares two values or expressions using symbols such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: There are several methods for solving inequalities, including graphing, factoring, and using algebraic methods. The method you choose will depend on the type of inequality and the values of the variable.

Q: What is the symbol for "less than or equal to"?

A: The symbol for "less than or equal to" is ≤.

Q: What is the symbol for "greater than or equal to"?

A: The symbol for "greater than or equal to" is ≥.

Q: How do I compare two fractions?

A: To compare two fractions, you can convert them to equivalent fractions with a common denominator. You can also compare the numerators and denominators separately.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that involves a linear expression, such as ${x + 2 > 3}$ . A quadratic inequality is an inequality that involves a quadratic expression, such as ${x^2 + 4x + 4 > 0}$ .

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use factoring, the quadratic formula, or graphing. The method you choose will depend on the type of inequality and the values of the variable.

Q: What is the difference between a polynomial inequality and a rational inequality?

A: A polynomial inequality is an inequality that involves a polynomial expression, such as ${x^3 + 2x^2 - 3x - 1 > 0}$ . A rational inequality is an inequality that involves a rational expression, such as ${\dfrac{x + 2}{x - 1} > 0}$ .

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you can use factoring, the quadratic formula, or graphing. The method you choose will depend on the type of inequality and the values of the variable.

Q: What is the difference between a one-sided inequality and a two-sided inequality?

A: A one-sided inequality is an inequality that involves a single inequality symbol, such as ${x > 2}$ . A two-sided inequality is an inequality that involves two inequality symbols, such as ${x > 2}$ and ${x < 4}$ .

Q: How do I solve a one-sided inequality?

A: To solve a one-sided inequality, you can use graphing, factoring, or algebraic methods. The method you choose will depend on the type of inequality and the values of the variable.

Q: What is the difference between a linear programming problem and an inequality problem?

A: A linear programming problem is a problem that involves maximizing or minimizing a linear function subject to a set of linear constraints. An inequality problem is a problem that involves solving an inequality.

Q: How do I solve a linear programming problem?

A: To solve a linear programming problem, you can use the simplex method, the graphical method, or the dual simplex method. The method you choose will depend on the type of problem and the values of the variables.

Conclusion

In conclusion, inequalities are an essential concept in mathematics, and they have numerous applications in real-life situations. By understanding and solving inequalities, we can gain a deeper understanding of mathematical concepts and apply them to real-world problems.