Problem 22 - Algebra 1: Concepts and Skills

Question

Solve the inequality. $$ 4 x-1 \leq-17 $$

Step-by-Step Solution

Answer

The solution to the inequality '4x - 1 ≤ -17' is \( x ≤ -4 \)

Step 1: Add 1 to both sides of the inequality

To isolate the '4x' term on one side, add 1 to both sides of the inequality. This yields: \[4x - 1 + 1 ≤ -17 + 1 \rightarrow 4x ≤ -16 \]

Step 2: Divide both sides by 4

Next to isolate 'x', divide both sides of the inequality by 4. This operation gives: \[ \frac{4x}{4} ≤ \frac{-16}{4} \rightarrow x ≤ -4 \]

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or inequalities. It's a fundamental skill in algebra, helping to isolate variables and make equations easier to solve. In the process of solving the inequality \( 4x - 1 \leq -17 \), we employ algebraic manipulation to isolate \( x \):

*We start by adding 1 to both sides*. This balances the inequality and removes the constant term on the left side, leading to: \[ 4x - 1 + 1 \leq -17 + 1 \rightarrow 4x \leq -16 \]
This step is crucial because it keeps the inequality balanced, similar to maintaining equality in equations by performing the same operation on both sides. Once the constant is removed, we focus on the coefficient of \( x \).

Next, by dividing both sides by 4, which is the coefficient of \( x \), we isolate \( x \). Thus, the term simplifies as follows: \[ \frac{4x}{4} \leq \frac{-16}{4} \rightarrow x \leq -4 \]
This step serves the purpose of reducing the inequality to its simplest form, presenting the solution \( x \leq -4 \). Once \( x \) is isolated, we understand where it lies on the number line.

Inequality Properties

Inequalities describe relationships between values, indicating that one expression is greater than, less than, or equal to another. Understanding inequality properties is crucial in manipulating and solving these mathematical statements.

When solving inequalities, several key properties need to be considered:

• Transitive Property: If \( a \leq b \) and \( b \leq c \), then \( a \leq c \).
• Additive Property: Adding or subtracting the same number from both sides maintains the inequality.
• Multiplicative Property: Multiplying or dividing both sides by a positive number keeps the inequality direction the same. However, if you multiply or divide by a negative number, the inequality direction flips.

For the inequality \( 4x - 1 \leq -17 \), we primarily use the additive and multiplicative properties. *Adding 1* maintains the direction, leading to \( 4x \leq -16 \). Dividing by 4, a positive number, preserves the direction, resulting in \( x \leq -4 \). Understanding these properties ensures you handle inequalities correctly while solving.

Linear Inequalities

Linear inequalities are algebraic expressions that involve a linear polynomial on one or both sides. They express a range of possible solutions rather than a single answer, unlike linear equations. Solving them helps find a set of values that satisfy the condition of the inequality. In \( 4x - 1 \leq -17 \), we're dealing with an inequality involving the linear term \( 4x \).

To solve these:

• Isolate the variable on one side, using algebraic manipulation techniques like addition and division.
• Apply inequality properties, ensuring the solution remains valid.
• Represent the solution as a range or on a number line.

Once simplified to \( x \leq -4 \), this inequality describes all values of \( x \) that are less than or equal to -4.

Linear inequalities can describe constraints in real-world problems, highlighting their significance in fields like economics or engineering. By mastering linear inequalities, you gain the ability to handle a wide range of scenarios where limits or boundaries are important.