Problem 53 - Algebra 1

Question

Is the given number a solution of the inequality? $$-2-x \leq-9 ;-7$$

Step-by-Step Solution

Answer

No, -7 is not a solution to the inequality -2 - x ≤ -9.

Step 1: Substitute the given number in the inequality

Substitute -7 for x in the inequality -2 - x ≤ -9. This gives: -2 - (-7) ≤ -9, which simplifies to -2 + 7 ≤ -9.

Step 2: Simplify the inequality

After the simplification in step 1, we end up with 5 ≤ -9.

Step 3: Check the inequality

The inequality 5 ≤ -9 is not true, a larger number cannot be smaller or equal to a smaller number.

Key Concepts

Inequality Solution Check

Understanding if a particular number satisfies an inequality is essential in dealing with algebraic problems. This process is known as an inequality solution check. The main goal here is to verify whether after substituting a given number into the inequality, the statement remains true or becomes false.

To perform an inequality solution check, you simply substitute the given number into the variable's place and simplify the expression. In the context of the exercise provided, with the inequality \( -2-x \leq -9 \) and the number -7, you would replace 'x' with -7. The resulting statement after substitution and simplification would then be analyzed for its truth-value. If the resulting statement does not hold true—as in the case of concluding that 5 is not less than or equal to -9—then the number is not a solution to the inequality. This check is a straightforward method to test potential solutions and is especially useful when dealing with linear inequalities.

Substitution Method

The substitution method is a foundational tool in algebra and is particularly helpful when one is tasked with solving equations or inequalities. When a value for a variable is suggested, substitution allows us to replace the variable with its potential value in order to determine if it makes the inequality true or false.

While applying this method in inequalities, precision is particularly important. For the exercise \( -2-x \leq -9 \), the substitution method requires inserting -7 in place of the variable x. It's necessary to pay attention to the signs when performing the substitution; for instance, the minus sign before x turns the subsequent -7 into a positive when substituted because of the double negative resulting in \( -2 + 7 \leq -9 \). If done correctly, the substitution method can quickly and efficiently reveal whether the value in question is a solution to the inequality.

Inequality Simplification

In any given inequality, inequality simplification is the process of reducing the expressions on either side of the inequality sign to their simplest form. This involves performing arithmetic operations such as addition, subtraction, multiplication, or division as appropriate, while also being mindful of maintaining the inequality relationship.

In the given exercise, once the substitution of -7 is made, we end up with \( -2 + 7 \leq -9 \), which simplifies to \( 5 \leq -9 \). Simplification is crucial as it allows us to see the inequality in its most basic and clear state. Only after simplifying can you accurately determine the validity of the inequality; in this case, it’s clear that the inequality does not hold true after simplification. It is key to remember that when multiplying or dividing by negative numbers in inequalities, the inequality sign must be flipped to maintain the truth of the statement.