Problem 52 - Algebra and Trigonometry

Question

In Exercises 51–58, solve each compound inequality. $$ 7

Step-by-Step Solution

Answer

The solution to the compound inequality \(7

Step 1: Remove the number added to x

In this case, \(x\) is added by 5 in the compound inequality. To isolate \(x\), you will subtract 5 from all parts of the inequality. This results in:\[ 7 - 5 < x + 5 - 5 < 11 - 5 \]which simplifies to:\[ 2 < x < 6 \]

Step 2: Interpret the results

The result \(2 < x < 6\) means that \(x\) is greater than 2 and less than 6. This is your solution set.

Key Concepts

Inequalities

Inequalities are essential to understand because they describe a range of possible values, rather than one exact solution. When we look at a compound inequality, such as \(7 < x + 5 < 11\), we're examining two inequalities linked together. It's like saying 'x plus five is more than seven AND less than eleven.' The key to solving these types of problems lies in treating each part of the inequality separately while keeping them linked.

In the given compound inequality, we're working with two boundaries to define a set where the variable \(x\) must fall within. This means that the value for \(x\) is not fixed but can vary within a specific range. When you're solving these, remember to keep the inequalities 'balanced' by performing the same operation on every part. In this way, you methodically narrow down the range in which the variable lies.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They're the building blocks of algebra and are used to represent real-world quantities. The expression \(x + 5\) in the compound inequality is a simple algebraic expression involving the variable \(x\) and a constant.

When solving compound inequalities, understanding how to manipulate these expressions is crucial. Subtracting 5 from the expression \(x + 5\), we simplify the inequality without altering its meaning; it's a matter of breaking down the algebra into simpler terms. The manipulation of algebraic expressions involves performing operations that will help us 'isolate' the variable, bringing us a step closer to finding the solution set.

Isolate Variable

The goal of 'isolating the variable' is to get the variable by itself on one side of the inequality to understand its potential values. In the context of our compound inequality, we perform the same operation across every part to maintain the balance of the equation. By subtracting 5 from all three parts of \(7 < x + 5 < 11\), we simplify to \(2 < x < 6\).

The variable \(x\) is now 'isolated,' meaning it stands alone without any added numbers. This step is crucial as it provides clarity on the possible values for \(x\), making it easy to understand the range of solutions. Visualize isolating the variable as finding the space where the variable 'lives,' free from added complexity. With \(x\) successfully isolated, interpretation follows smoothly, guiding you to the interval in which the variable lies.

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Problem 51

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Problem 52

Other exercises in this chapter

Problem 51

Find all values of x satisfying the given conditions. $$y_{1}=5(2 x-8)-2, y_{2}=5(x-3)+3, \text { and } y_{1}=y_{2}$$

View solution

Problem 51

Evaluate \(x^{2}-2 x+2\) for \(x=1+i\)

View solution

Problem 52

Solve equation by completing the square. $$ x^{2}-2 x-5=0 $$

View solution

Problem 52

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x^{\frac{2}{5}}+x^{\frac{1}{5}}-6=0$$

View solution

Key Concepts

Inequalities

Inequalities are essential to understand because they describe a range of possible values, rather than one exact solution. When we look at a compound inequality, such as \(7 < x + 5 < 11\), we're examining two inequalities linked together. It's like saying 'x plus five is more than seven AND less than eleven.' The key to solving these types of problems lies in treating each part of the inequality separately while keeping them linked.

In the given compound inequality, we're working with two boundaries to define a set where the variable \(x\) must fall within. This means that the value for \(x\) is not fixed but can vary within a specific range. When you're solving these, remember to keep the inequalities 'balanced' by performing the same operation on every part. In this way, you methodically narrow down the range in which the variable lies.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They're the building blocks of algebra and are used to represent real-world quantities. The expression \(x + 5\) in the compound inequality is a simple algebraic expression involving the variable \(x\) and a constant.

When solving compound inequalities, understanding how to manipulate these expressions is crucial. Subtracting 5 from the expression \(x + 5\), we simplify the inequality without altering its meaning; it's a matter of breaking down the algebra into simpler terms. The manipulation of algebraic expressions involves performing operations that will help us 'isolate' the variable, bringing us a step closer to finding the solution set.

Isolate Variable

The goal of 'isolating the variable' is to get the variable by itself on one side of the inequality to understand its potential values. In the context of our compound inequality, we perform the same operation across every part to maintain the balance of the equation. By subtracting 5 from all three parts of \(7 < x + 5 < 11\), we simplify to \(2 < x < 6\).

The variable \(x\) is now 'isolated,' meaning it stands alone without any added numbers. This step is crucial as it provides clarity on the possible values for \(x\), making it easy to understand the range of solutions. Visualize isolating the variable as finding the space where the variable 'lives,' free from added complexity. With \(x\) successfully isolated, interpretation follows smoothly, guiding you to the interval in which the variable lies.