Problem 57 - Algebra 1

Question

Check whether the number is a solution of the equation or the inequality. \(s-7 \geq 12-s ; 9\)

Step-by-Step Solution

Answer

No, the number 9 is not a solution for the given inequality since the inequality \(2 \geq 3\) is not true.

Step 1 - Substitute the given number into the inequality

Replace the 's' variable by '9' in the inequality: \(9-7 \geq 12-9\)

Step 2 - Perform the operations

When we perform the subtraction on both sides of inequality, we get: \(2 \geq 3\)

Step 3 - Check the validity of the inequality

To determine if '9' is a solution for the inequality, we need to check if the statement \(2 \geq 3\) is valid or not.

Key Concepts

Inequality Solutions

Inequalities in algebra represent relationships where one side is not exactly equal to another, but rather greater than or less than it. Unlike equations, which have a distinct solution, inequalities provide a range of possible values that satisfy the condition given. For instance, in the inequality \(s - 7 \geq 12 - s\), not just one number makes the inequality true. In this example, you are tasked with figuring out if the number '9' makes the inequality true.

• First, substitute '9' in place of 's'.
• Then, solve both sides of the inequality to see if the left side is indeed greater than or equal to the right side.

Inequalities can be tricky at first but practicing substitution, solving, and verification can help you master this concept.
Remember: inequalities can also have many solutions or even none, depending on the context.

Check Validity

Checking validity is an essential step in solving inequalities, as it ensures that the proposed solution is correct. After substituting the given value into the inequality, as we did with the number '9', it's crucial to evaluate the outcome.Here’s how to check validity:

• Perform arithmetic operations on both sides of the inequality.
• Compare the results to verify if the inequality holds true.

For our example, substituting '9' resulted in the inequality \(2 \geq 3\). Since 2 is not greater than or equal to 3, the statement is false.
Hence, '9' is not a valid solution for this inequality.
Checking validity helps you certify that only true solutions are accepted and highlighted.

Substitution Method

The substitution method is a simple and effective technique used to test specific values within an equation or inequality. It involves replacing variables with actual numbers to see if the statement becomes true.To apply the substitution method:

• Identify the variable in the inequality.
• Replace the variable with the given number.

For instance, in \(s - 7 \geq 12 - s\), 's' is replaced with '9'. This allows you to perform arithmetic to find out whether the statement holds. After substitution:

• Carry out the calculations on both sides to simplify.
• Evaluate the final expressions to check if the inequality is true.

Using substitution is beneficial for testing specific numbers quickly without solving the inequality entirely. It is especially advantageous for checking potential solutions when you are given particular numbers."}]}]}

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Other exercises in this chapter

Problem 57

Evaluate the expression for the given value of the variable. $$ 7 b^{2} \text { when } b=7 $$

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Problem 57

$$ \text {} 8 \frac{9}{10}+1 \frac{1}{5} $$

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Problem 58

Evaluate the expression for the given value of the variable. $$ 2 x^{2} \text { when } x=15 $$

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Problem 58

EQUATIONS AND INEQUALITIES Match the verbal sentence with its mathematical representation. The difference of \(x\) and 16 is 32

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