Problem 5 - Elementary Algebra

Question

Determine whether the given number is a solution to the given inequality. $$ -5<7 x+1<9 ; x=0 $$

Step-by-Step Solution

Answer

Yes, x=0 is a solution to the inequality.

Step 1: Substitute the given value of x into the inequality

We start by substituting the given value \( x = 0 \) into the inequality \( -5 < 7x + 1 < 9 \). This means replacing \( x \) with 0.

Step 2: Simplify the inequality

Substitute \( x = 0 \) into \( 7x + 1 \), we get: \( 7(0) + 1 = 0 + 1 = 1 \). So we have to check if \( -5 < 1 < 9 \).

Step 3: Test whether the simplified expression satisfies the inequality

Now verify if \( 1 \) satisfies the inequality \( -5 < 1 < 9 \). This means checking if \( 1 \) is greater than \(-5\) and less than \(9\).

Step 4: Conclusion

Since \( 1 \) is indeed greater than \(-5\) and less than \(9\), the inequality \(-5 < 1 < 9\) holds true. Therefore, \( x = 0 \) is a solution to the inequality.

Key Concepts

Solution Verification

Verifying a solution to an inequality is an essential skill in algebra. It involves determining whether a given value indeed satisfies the specified conditions of an inequality. To verify a solution, follow these steps:

• Substitute the given value into the inequality equation.
• Simplify the resulting expressions.
• Check if the simplified values meet all inequality conditions.

In the exercise, we verified if \( x = 0 \) is a valid solution for the inequality \(-5 < 7x + 1 < 9\). By substituting \( x = 0 \) into the expression, we simplified it to \(-5 < 1 < 9\). Since 1 fits within the range given in the inequality, we confirmed that \( x = 0 \) is indeed a solution. This thorough checking process is essential in mathematics, as it ensures that the solution fully complies with the inequality's requirements.

Substitution Method

The substitution method helps us check if a specific value satisfies an equation or an inequality. It involves replacing the variable with its given value. Here’s how you can effectively use substitution:

• Identify the variable and its given value.
• Replace every occurrence of the variable in the expression with the given value.
• Simplify the expression to see its resulting value.

In the problem provided, we substituted \( x = 0 \) into the compound inequality \(-5 < 7x + 1 < 9\). This means substituting \( x \) with 0 to obtain \( 7(0) + 1 \). Performing the arithmetic simplification yields 1, at which point the inequality becomes \(-5 < 1 < 9\). This type of straightforward substitution allows us to determine quickly whether the provided value satisfies the inequality criteria.

Algebraic Expressions

Algebraic expressions form the backbone of many mathematical problems, including inequalities. They consist of numbers, variables, and arithmetic operations. Understanding how to manipulate algebraic expressions is crucial.When dealing with algebraic expressions in inequalities:

• Pay attention to the order of operations to ensure accurate simplification.
• Perform operations such as addition and multiplication step by step, especially for compound expressions.
• Check each part of a compound inequality separately, ensuring the entire range is satisfied.

In our example, the algebraic expression \(7x + 1\) was central to the inequality \(-5 < 7x + 1 < 9\). Accurately substituting and evaluating this expression allowed us to verify if \( x = 0 \) made the inequality true. Therefore, mastering the handling of algebraic expressions is essential for solving such problems effectively.