Problem 89 - Introductory Algebra for College Students

Question

Solve each inequality. \(5 x-4 \leq 4(x-1)\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq 0\).

Step 1: Distribute on the right side

The right side of the inequality \(4(x-1)\) contains a parenthesis. To get rid of the parenthesis, distribute by multiplying every term inside by 4: \(4*x - 4*1 \) which simplifies to \(4x - 4\).

Step 2: Replace the right side

After distributing, the inequality now reads: \(5x - 4 \leq 4x - 4 \)

Step 3: Isolate x terms on one side

Subtract \(4x\) from both sides of the inequality: \(5x - 4x \leq 4x - 4x\) which simplifies to \(x - 4 \leq -4\).

Step 4: Isolate x

Get x alone on one side by adding 4 to both sides: \(x - 4 + 4 \leq -4 + 4\) which results to \(x \leq 0\).

Key Concepts

Algebraic Inequalities

Understanding algebraic inequalities is crucial for solving many types of algebra problems. An inequality, as opposed to an equation, indicates that one expression is greater or lesser than another, rather than being equal. Types of inequalities include '\textless', '\textgreater', '\textless=', and '\textgreater='. When solving inequalities, much like equations, you aim to isolate the variable on one side to determine its range of values that satisfy the inequality. However, a unique feature of solving inequalities is the need to reverse the inequality sign when both sides are multiplied or divided by a negative number.

In the context of our example, we were solving the inequality \(5x - 4 \textless= 4(x - 1)\). Here, we seek to find all values of 'x' that would make the inequality true. By performing a series of steps that maintain the balance of the inequality—similar to solving equations—we discover that \(x \textless= 0\) is the solution, meaning that any 'x' value that is zero or negative will satisfy the original inequality.

Inequality Notation

Inequality notation is the system used to represent inequalities symbolically. The symbol '\textless=' means 'less than or equal to', and '\textgreater=' means 'greater than or equal to'. These are crucial for understanding the range of solutions to an inequality. They tell us not just about a single number, but about a whole set of numbers that can work as solutions. For example, the solution to the given inequality \(x \textless= 0\), with the '\textless=' sign demonstrates that the solution is not just zero, but all real numbers less than or equal to zero. This is a crucial concept in algebra since it highlights that often we're looking for a range rather than a single value. Proper use of inequality notation is essential in correctly conveying the solution set for an inequality.

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify expressions and solve equations and inequalities efficiently. It states that for any numbers or expressions a, b, and c, \(a \times (b + c) = ab + ac\). This property is instrumental in the simplification process within inequalities.

Looking at the given exercise where we need to solve \(5x - 4 \textless= 4(x - 1)\), we apply the distributive property in step 1 to eliminate the parentheses by distributing the 4 to both x and -1. This gives us \(4x - 4\) on the right side. This step is crucial because it breaks down the expression into a simpler form without changing the inequality's value or solution. The distributive property is a powerful tool in continually reducing and simplifying until we reach the simplest form that can be easily solved or understood.