Problem 40 - Algebra 1

Question

Decide whether the ordered pair is a solution of the inequality. $$y

Step-by-Step Solution

Answer

No, the ordered pair (1,1) is not a solution to the inequality.

Step 1: Substitute the values into the inequality

The ordered pair provided is (1,1). This should be substituted into the inequality, i.e., \(y

Step 2: Solve the inequality

Having substituted the values, resolve the right side of the inequality. This is achieved by simplifying \(1^{2}-2*1-5\), which equals -4. Therefore, the inequality now reads \(1<-4\).

Step 3: Decide if the inequality is true

Looking at the inequality \(1<-4\), it is clear that 1 is not less than -4. Thus, the inequality is false, meaning the ordered pair (1,1) is not a solution.

Key Concepts

Ordered Pair

An ordered pair consists of two elements, typically written as \((x, y)\), where \(x\) is the first component and \(y\) is the second.It is used to represent a point in the two-dimensional coordinate plane.Ordered pairs are fundamental in representing relationships in mathematics, particularly in graphing equations.In the context of inequalities, an ordered pair can be evaluated to determine whether it satisfies the given inequality.This involves substituting the values of the ordered pair into the inequality.If the resulting expression forms a true statement, then the ordered pair is indeed a solution.Otherwise, it is not.Understanding the concept of ordered pairs is crucial when working with coordinate systems and analyzing solutions to various mathematical problems.Whenever you see an ordered pair, remember it represents a specific point on the graph.

Solution of Inequality

To determine if an ordered pair is a solution of an inequality, the components of the pair are substituted into the inequality.Once substituted, both sides of the inequality are evaluated and compared.Consider the inequality given in the exercise: \( y < x^2 - 2x - 5 \).We substitute \((x, y) = (1,1)\) into the inequality:

• Replace \(x\) with 1 and \(y\) with 1
• Evaluate: \( 1 < 1^2 - 2 \times 1 - 5 \)
• Simplify the right side: \( 1 < -4 \)

As you can see, \( 1 < -4 \) is not true, so the ordered pair \((1, 1)\) is not a solution.Being a solution means that once the ordered pair is substituted into the inequality, the statement holds true.If it does not, the pair fails as a solution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations.In the inequality \( y < x^2 - 2x - 5 \), the expression on the right side, \( x^2 - 2x - 5 \), is an algebraic expression.Key features of algebraic expressions include:

• Variables: Symbols such as \(x\), \(y\), and others are used to represent unknown values.
• Operations: Including addition, subtraction, multiplication, and division, which are used to form the expression.
• Constants: Numbers like \(-5\) that add fixed values to the expression.

This expression, \( x^2 - 2x - 5 \), is evaluated by substituting specific values for the variable \(x\).Once the variable \(x\) from the ordered pair \((1, 1)\) is substituted, the expression simplifies to \(-4\). The inequality then reads \(1 < -4\), which is a false statement in this situation.Understanding how to work with and manipulate algebraic expressions is a central skill in solving equations and inequalities. It allows you to rigorously evaluate mathematical statements.

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Problem 40

Other exercises in this chapter

Problem 40

Solve the equation. Check for extraneous solutions. $$x=\sqrt{11 x-28}$$

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Problem 40

Find the midpoint between the two points \((-1,1),(-4,-4)\)

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Problem 40

SIMPLIFYING RATIONAL EXPRESSIONS Simplify the expression. $$\frac{3}{x}+\frac{x+9}{x}$$

View solution

Problem 40

Find the domain of the function. $$y=x \sqrt{x}$$

View solution

Key Concepts

Ordered Pair

An ordered pair consists of two elements, typically written as \((x, y)\), where \(x\) is the first component and \(y\) is the second.It is used to represent a point in the two-dimensional coordinate plane.Ordered pairs are fundamental in representing relationships in mathematics, particularly in graphing equations.In the context of inequalities, an ordered pair can be evaluated to determine whether it satisfies the given inequality.This involves substituting the values of the ordered pair into the inequality.If the resulting expression forms a true statement, then the ordered pair is indeed a solution.Otherwise, it is not.Understanding the concept of ordered pairs is crucial when working with coordinate systems and analyzing solutions to various mathematical problems.Whenever you see an ordered pair, remember it represents a specific point on the graph.

Solution of Inequality

To determine if an ordered pair is a solution of an inequality, the components of the pair are substituted into the inequality.Once substituted, both sides of the inequality are evaluated and compared.Consider the inequality given in the exercise: \( y < x^2 - 2x - 5 \).We substitute \((x, y) = (1,1)\) into the inequality:

• Replace \(x\) with 1 and \(y\) with 1
• Evaluate: \( 1 < 1^2 - 2 \times 1 - 5 \)
• Simplify the right side: \( 1 < -4 \)

As you can see, \( 1 < -4 \) is not true, so the ordered pair \((1, 1)\) is not a solution.Being a solution means that once the ordered pair is substituted into the inequality, the statement holds true.If it does not, the pair fails as a solution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations.In the inequality \( y < x^2 - 2x - 5 \), the expression on the right side, \( x^2 - 2x - 5 \), is an algebraic expression.Key features of algebraic expressions include:

• Variables: Symbols such as \(x\), \(y\), and others are used to represent unknown values.
• Operations: Including addition, subtraction, multiplication, and division, which are used to form the expression.
• Constants: Numbers like \(-5\) that add fixed values to the expression.

This expression, \( x^2 - 2x - 5 \), is evaluated by substituting specific values for the variable \(x\).Once the variable \(x\) from the ordered pair \((1, 1)\) is substituted, the expression simplifies to \(-4\). The inequality then reads \(1 < -4\), which is a false statement in this situation.Understanding how to work with and manipulate algebraic expressions is a central skill in solving equations and inequalities. It allows you to rigorously evaluate mathematical statements.