Problem 55 - Algebra 1: Concepts and Skills

Question

Check whether the given value of the variable is a solution of the inequality. (Lesson 1.4) $$ 12 a \leq a-9 ; a=-2 $$

Step-by-Step Solution

Answer

-2 is a solution of the inequality \(12a \leq a - 9\).

Step 1: Understand the inequality

Observe an inequality \(12a \leq a - 9\). It claims that a product of 12 and some number \(a\) is less or equal than the same number \(a\) decreased by 9.

Step 2: Substitute the given value

The given value for \(a\) is -2. So, replace \(a\) with -2 in the inequality \(12a \leq a - 9\) to get \(12(-2) \leq (-2) -9\). After doing the multiplication and subtraction, the inequality becomes \(-24 \leq -11\)

Step 3: Verify the inequality

Now check the inequality \(-24 \leq -11\). This inequality states that -24 is less or equal to -11. Since -24 is indeed less than -11, the inequality is true.

Key Concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions that show the relationship between two values where one is not strictly equal to the other but could be either less than, greater than, or equal whilst also being less or greater. They play a vital role in representing the constraints and limitations in real-world situations, ranging from simple calculations to complex problem-solving.

For example, the inequality from the exercise, \(12a \leq a - 9\), can be interpreted as a statement that the value of \(12a\) is always less than or equal to the value of \(a\) subtracted by 9. In real life, this could represent a scenario where the capacity of a container (represented by \(12a\)) is limited to be less than or equal to the volume of substance it can hold after losing a portion.

Substitution Method

The substitution method is a technique used to solve systems of equations, and it can also be applied to inequalities. The method involves replacing a variable with its given or calculated value to simplify the equation or inequality.

Using the exercise as an example, you're provided with the value of \(a = -2\). Substituting this into the original inequality \(12a \leq a - 9\) gives us \(12(-2) \leq (-2) - 9\), transforming the algebraic expression into a numerical one that is easier to comprehend and solve. This crucial step transforms the abstract inequality into a concrete statement that can then be verified or refuted.

Inequality Verification

Inequality verification is the process of confirming whether a given number satisfies the inequality in question. After substituting the variable in the inequality with the given value, you arrive at a statement that can be judged as true or false. This final step ensures that the solution complies with the original condition.

In our example, the inequality after substitution becomes \(-24 \leq -11\). In the context of number lines, -24 is to the left of -11, which agrees with the direction of the inequality symbol \(\leq\). Therefore, the verification process concludes that -24 is indeed less than or equal to -11, and as such, \(a = -2\) is a valid solution to the initial inequality. This is an illustrative example of how verification is used to ensure that the solution not only makes mathematical sense but also aligns with the logical structure of inequalities.