Problem 52 - Prealgebra

Question

ALGEBRA For the given value, state whether each inequality is true or false. (Lesson 8-3) $$9+a \leq 3, a=-7$$

Step-by-Step Solution

Answer

The inequality is true.

Step 1: Substitute the Value of a

Start by substituting the given value of \( a = -7 \) into the inequality. The original inequality is \( 9 + a \leq 3 \). By substituting \( a \), it becomes \( 9 + (-7) \leq 3 \).

Step 2: Simplify the Expression

Simplify the expression on the left side. Calculate the sum of 9 and -7: \( 9 + (-7) = 2 \). So the inequality now is \( 2 \leq 3 \).

Step 3: Evaluate the Inequality

Determine whether the inequality \( 2 \leq 3 \) is true or false. Since 2 is indeed less than or equal to 3, the inequality is true.

Key Concepts

Understanding Algebra

Algebra is a fascinating field in mathematics that deals with symbols and the rules for manipulating these symbols to express mathematical relationships. These symbols often represent numbers in equations and inequalities. In algebra, you sometimes solve for unknown values represented by variables like \( a \) or \( x \). This is essential for forming and solving equations.
Understanding algebra helps you to:

• Create equations and inequalities to model real-world situations.
• Work with unknowns that vary and find their possible values.

When tackling an inequality, similar to an equation, the goal is to find the value or range of values for the variable that make the inequality true. Inequalities are expressions demonstrating a relationship where one side is not necessarily equal to the other, often represented using symbols like \( \leq \), \( \geq \), \( < \), and \( > \). Learning algebra provides the foundational skills needed to work with these mathematical statements and understand their overarching principles.

Mastering Substitution

Substitution is a powerful tool in algebra that makes solving equations and inequalities more accessible. It involves replacing a variable with a specific value or another expression. This simplification technique allows you to transform an abstract problem into a concrete numeric one. For instance, if you know \( a = -7 \), you're substituting \( a \) with \( -7 \) in the expression.Here's why substitution is essential:

• It lets you replace symbols with known values for computation.
• It helps simplify complex expressions into manageable calculations.

In our inequality example, substituting the value of \( a \) is the first critical step. By replacing \( a \) with \( -7 \), the abstract inequality \( 9 + a \leq 3 \) becomes \( 9 + (-7) \leq 3 \), transforming it into a problem with numbers instead of variables.

Simplification Techniques

Simplification in algebra involves reducing expressions to their simplest form while retaining their original value or truth. This process often comprises several sub-steps: combining like terms, performing arithmetic operations, and canceling out terms when possible, to make expressions cleaner and solvable. In the context of our inequality, simplification focuses on simplifying both sides of the inequality after substitution.
After substituting \( a = -7 \):

• The expression \( 9 + (-7) \) is simplified to \( 2 \).
• This is straightforward because it involves basic addition of positive and negative numbers.

Simplifying makes it easier to visually see relationships and identify solutions without overly complicated calculations. It is a vital step because simpler expressions are more easily interpreted, allowing for quick evaluations of inequalities or equations.

True or False Evaluation

Once an algebraic expression is simplified, the next step is to evaluate whether the resulting statement is true or false. This process involves understanding the basic logical operators used in inequalities. In our example, you ended with the inequality \( 2 \leq 3 \). Knowing:

• \( \leq \) means less than or equal to.
• Evaluating \( 2 \leq 3 \) involves checking if 2 is indeed less than or equal to 3.

As \( 2 \) is less than \( 3 \), the inequality is true. This final step, evaluating the simplified expression, is straightforward because it uses basic number comparison. Understanding the properties of inequalities and practicing comparison helps ensure accurate conclusions, reinforcing fundamental arithmetic understanding essential for more advanced algebraic problem-solving.