Problem 16 - Elementary Algebra Within Reach

Question

Determine whether each value of \(x\) is a solution of the inequality. \(4-3 x \geq x+12\) (a) \(x=0\) (b) \(x=3\) (c) \(x=-2\) (d) \(x=-4\)

Step-by-Step Solution

Answer

The values \(x=-4\) and \(x=-2\) are solutions to the inequality.

Step 1: Substitute x=0

Substitute \(x = 0\) into the inequality, yielding \(4 - 3(0) \geq 0 + 12\). Simplifying that gives \(4 \geq 12\), which is false. Thus, \(x=0\) is not a solution.

Step 2: Substitute x=3

Substitute \(x = 3\) into the inequality, yielding \(4 - 3(3) \geq 3 + 12\). Simplifying that gives \(-5 \geq 15\), which is false. Thus, \(x=3\) is not a solution.

Step 3: Substitute x=-2

Substitute \(x = -2\) into the inequality, yielding \(4 - 3(-2) \geq -2 + 12\). Simplifying that gives \(10 \geq 10\), which is true. Thus, \(x=-2\) is a solution.

Step 4: Substitute x=-4

Substitute \(x = -4\) into the inequality, yielding \(4 - 3(-4) \geq -4 + 12\). Simplifying gives \(16 \geq 8\), which is true. Thus, \(x=-4\) is a solution.

Key Concepts

Inequality Solutions

When solving inequalities, the goal is to find the set of values that make the inequality true. Inequalities similar to equations can involve expressions with variables, like the inequality in our example: \(4-3x \geq x+12\). The difference between equations and inequalities is that inequalities use symbols like \(>\), \(<\), \(\geq\), and \(\leq\), instead of the equal sign \(=\). To verify if a particular value is a solution to an inequality:

• Substitute the value for the variable.
• Simplify both sides of the inequality.
• Check if the inequality statement holds true.

In our example, if the left side is greater than or equal to the right side after simplifying, then the chosen value for \(x\) is a solution. Otherwise, it is not.

Substitution Method

The substitution method is a straightforward approach used to determine whether a given number is a solution to an equation or inequality. Here’s how it works:Start by replacing the variable in the inequality with the given number. Simplify both sides of the inequality as much as possible. Observe whether the resulting statement is true or false.For instance, substituting \(x = 0\) in our inequality gives us \(4 - 3(0) \geq 0 + 12\), which simplifies to \(4 \geq 12\). Since this statement is false, \(x = 0\) is not a solution. On the contrary, substituting \(x = -2\) into the inequality results in a true statement: \(10 \geq 10\). Therefore, \(x = -2\) is indeed a solution. Utilizing substitution provides a clear, uncomplicated way to check different values quickly.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division) that represent a mathematical relationship. Understanding these expressions is essential when dealing with inequalities as they form the basis of both sides of the inequality. In our given example, the expressions \(4 - 3x\) and \(x + 12\) each represent one side of the inequality. By working with algebraic expressions, you can manipulate and simplify them to gain insight into the relationships they show.Here are a few tips on handling algebraic expressions:

• Identify like terms and combine them to simplify an expression.
• Use the distributive property to remove parentheses when necessary.
• Remember that simplifying expressions while respecting the order of operations is key to solving inequalities accurately.

These expressions and their simplifications are instrumental in solving problems like our inequality, making them a cornerstone of basic algebra.