Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$11 x \leq x-7 ; 9$$

Short answer

No, '9' is not a solution to the inequality \(11x \leq x - 7\).

Step-by-step solution

Write down the inequality

First, note down the given inequality: \(11x \leq x - 7\)

Substitute the given number for x

Substitute '9' for 'x' in the inequality and evaluate each side. So, we have \(11*9 \leq 9 - 7\). This simplifies to \(99 \leq 2\).

Check if the inequality is true

In the final step, check if the simplification in Step 2 is true. Here, it's clearly false because \(99\) is not less than or equal to \(2\).

Key concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions involving variables that show the relationship between two values, indicating that one is greater than, less than, or equal to the other. For example, in the inequality \(11x \leq x - 7\), \(x\) represents a variable, which can be any number that makes the inequality true. An inequality looks much like an equation but instead of an equal sign \(=\), it uses signs like \(>\), \(<\), \(\geq\), or \(\leq\) to relate the expressions on either side.

Understanding inequalities is critical as they are used to describe a range of possible answers in real-world problems, such as minimum or maximum limits. Instead of looking for a single solution, as in an equation, an inequality often has a set of solutions that satisfy it. Therefore, it's important to grasp how to manipulate and solve these inequalities by applying algebraic rules while preserving the direction and sense of the inequality.

Substitution Method

The substitution method is a quintessential technique used to solve algebraic inequalities, as well as equations. It involves replacing a variable in an inequality with a given or known value to determine if the inequality holds true. For example, in our exercise, the inequality \(11x \leq x - 7\) can be evaluated for a particular value of \(x\) by substituting that value into the inequality. If we want to see if \(9\) is a solution, we use the substitution method by replacing \(x\) with \(9\) and then evaluating the resulting expression.

This is a practical approach when you have a suspected solution you wish to verify. Despite its simplicity, it requires careful arithmetic to ensure accurate substitution and evaluation, as errors in this fundamental step can lead to incorrect conclusions about the solutions of the inequality.

Inequality Evaluation

Inequality evaluation is the process of determining the truthfulness of the inequality after a value is substituted into the variable. The process begins once a value has been substituted into the inequality, which is then simplified. In our provided example, after substituting \(9\) into the inequality to get \(11 \times 9 \leq 9 - 7\), we then evaluate whether \(99 \leq 2\) holds true.

We can clearly see that \(99\) is not less than or equal to \(2\), and so we conclude that \(9\) is not a solution to the inequality \(11x \leq x - 7\). This step is vital as it confirms the validity of a solution. Understanding this process is necessary to solve real-life problems where certain constraints must be satisfied, and this method helps establish whether a particular choice adheres to those constraints.