Problem 21 - Prealgebra

Question

Solve each inequality. Check your solution. Then graph the solution on a number line. $$-8 \leq-4 w$$

Step-by-Step Solution

Answer

The solution is \(w \leq 2\). Graph the solution with a closed circle at 2 and shade left.

Step 1: Isolate the Variable

To solve the inequality \(-8 \leq -4w\), we need to isolate \(w\). Divide each side by \(-4\) to obtain the inequality: \(2 \geq w\). Remember, when you divide or multiply both sides of an inequality by a negative number, the inequality sign flips.

Step 2: Write the Solution

Now that we have simplified the inequality to \(w \leq 2\), we can express this as the solution: \(w \leq 2\).

Step 3: Check the Solution

To verify, pick a value less than or equal to 2 (e.g., \(w = 2\) and \(w = 0\)) and substitute it back into the original inequality. For \(w = 2\), \(-8 \leq -4(2)\) simplifies to \(-8 \leq -8\), which is true. For \(w = 0\), \(-8 \leq 0\) is also true, confirming the solution.

Step 4: Graph the Solution

To graph the inequality \(w \leq 2\) on a number line, draw a number line with points marking integer values. Place a closed circle on 2 (since 2 is included in the solution) and shade the line to the left of 2, indicating that all values less than or equal to 2 satisfy the inequality.

Key Concepts

Isolate the Variable

Facing an inequality like \(-8 \leq -4w\), our first mission is to isolate the variable \(w\). Isolating the variable means rearranging the equation so that \(w\) stands alone on one side. In this inequality, \(-4w\) needs to be by itself. To do this, we divide both sides by \(-4\). This step is crucial, as dividing by a negative number requires us to flip the inequality sign.
Thus, the inequality becomes \(2 \geq w\). Now, \(w\) is isolated and our inequality is much simpler: \(w \leq 2\). Throughout this process, attention to reversing the inequality sign ensures the solution accurately reflects the relationships expressed in the original problem.

Checking Solutions

After finding a potential solution, it’s always wise to check if it’s correct. For the inequality \(w \leq 2\), choose numbers that are equal to or less than 2. This gives us choices like \(w = 2\) and \(w = 0\).

• For \(w = 2\), substituting back into the original inequality gives us \(-8 \leq -8\), which holds true.
• For \(w = 0\), substituting back gives \(-8 \leq 0\), which also holds true.

Both substitutions confirm that our solution is correct. Checking solutions helps us verify that the isolated variable solution satisfies the true nature of the inequality and ensures there are no missteps.

Graphing on a Number Line

Visualizing solutions on a number line provides a clear picture of where the solutions of an inequality lie. For \(w \leq 2\), start by drawing a horizontal line representing numbers. Mark integers for clarity, especially near the value of interest, \(2\).
Place a closed circle over the number 2. A closed circle shows that the value 2 is included in the solutions, aligning with our "less than or equal to" condition.
Finally, shade the number line extending to the left of this point, indicating all numbers to the left, including 2, are part of the solution set.
This graph effectively communicates every possible value of \(w\) that satisfies the inequality \(w \leq 2\).