Problem 89 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

There are no real solutions.

Step 1: Expand the Brackets

The first step to solve this inequality is to expand the brackets on the right side of the inequality, thus giving us: \(5x - 4 \leq 4x - 4\).

Step 2: Rearrange the Inequality

The next step is to rearrange the inequality to isolate x on one side. To do so, we subtract \(4x\) from both sides, yielding: \(5x - 4x \leq - 4\), and simplifying, we get \(x \leq -4\).

Step 3: Check the Solution

The final step is to check the solution to ensure it makes the original inequality true. Plugging \(-4\) back into the original inequality gives \(5*(-4) - 4 \leq 4*(-4 - 1)\), which simplifies to \(-24 \leq -20\). Since this is not true, there are no real solutions to this inequality.

Key Concepts

Expand Brackets

Expanding brackets is an essential skill in algebra that helps to simplify expressions. When we expand brackets, we distribute the number outside the bracket to each term inside the bracket. This step often sets the stage for isolating the variable we're solving for. In our example, the inequality given is:\[5x - 4 \leq 4(x-1)\]To expand the brackets on the right side, we distribute the 4 to both the \(x\) and the \(-1\). This results in:\[4(x-1) = 4x - 4\]After expanding, the inequality becomes:\[5x - 4 \leq 4x - 4\]This simplifies our equation, allowing us to see the relationships between the terms more clearly. Properly expanding brackets is critical as it lays the groundwork for any further manipulation of the inequality.

Rearrange Inequality

Rearranging an inequality involves moving terms around so that the variable, in this case, \(x\), is isolated on one side. This helps us determine the possible values of \(x\) that satisfy the inequality. Starting with our expanded inequality:\[5x - 4 \leq 4x - 4\]We want the \(x\) terms to appear on one side. To achieve this, subtract \(4x\) from both sides:\[5x - 4x - 4 \leq -4\]Next, simplify the left side by combining like terms:\[x - 4 \leq -4\]Having \(x\) alone on one side of the inequality lets us directly read possible solutions. It simplifies to:\[x \leq 0\]Rearranging the inequality is crucial because it clarifies the relationship between the terms, making it easier to understand what values of \(x\) are possible solutions.

Check Solution

Checking the solution is a vital last step when solving inequalities. This confirms the possible solutions found indeed satisfy the original inequality.In this exercise, we've determined that \(x \leq 0\). Now, it’s time to verify by considering logical test points.Let's try plugging a test value like \(x = -1\) back into the original inequality:\[5(-1) - 4 \leq 4(-1 - 1)\]Simplifying, the left side yields:\[-5 - 4 = -9\]And the right side results in:\[4(-2) = -8\]Comparing \(-9 \leq -8\), we see this doesn't hold true. Since no positive result emerges after testing points within the identified range, this implies our inequality had no solutions.This checking process assures us that the potential solutions suggested by our inequality manipulation are valid or need revisiting.