Problem 20 - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line. $$ |3 w+2| \leq 5 $$

Step-by-Step Solution

Answer

The solution set is \(-\frac{7}{3} \leq w \leq 1\).

Step 1: Understand Absolute Value Inequality

An inequality of the form \(|x| \leq a\) means that \(-a \leq x \leq a\). This is because absolute value expresses distance from zero on a number line, so \(x\) must lie within \(a\) units of zero.

Step 2: Apply Absolute Value Rule to the Problem

Given \(|3w + 2| \leq 5\), we write it as two inequalities: \(-5 \leq 3w + 2 \leq 5\). This means the expression inside the absolute value, \(3w + 2\), must lie between \(-5\) and \(5\).

Step 3: Solve the Left Inequality

Solve \(-5 \leq 3w + 2\) for \(w\).\[-5 \leq 3w + 2\]Subtract 2 from both sides:\[-7 \leq 3w\]Divide each side by 3:\[-\frac{7}{3} \leq w\]This is the first part of the solution.

Step 4: Solve the Right Inequality

Solve \(3w + 2 \leq 5\) for \(w\).\[3w + 2 \leq 5\]Subtract 2 from both sides:\[3w \leq 3\]Divide each side by 3:\[w \leq 1\]This is the second part of the solution.

Step 5: Combine Both Inequalities

Combine the results from the previous steps, you get:\[-\frac{7}{3} \leq w \leq 1\]This represents the solution to the inequality. The value of \(w\) must be between \(-\frac{7}{3}\) and \(1\), inclusive.

Step 6: Graph the Solution on a Number Line

On a number line, mark the points \(-\frac{7}{3}\) and \(1\). Draw a closed circle at \(-\frac{7}{3}\) and at \(1\), because these points are included in the solution (indicated by the "less than or equal to" inequality). Shade the region between them, representing all values \(w\) between and including \(-\frac{7}{3}\) and \(1\).

Key Concepts

Inequality Solving

Solving inequalities is quite similar to solving equations, but with a few extra considerations. When working with inequalities, our goal is to find all possible values that satisfy the inequality. The trickiest part involves the rules of operations, especially when multiplying or dividing by negative numbers, which can reverse the inequality sign.
For the example given, \(|3w + 2| \leq 5\), we break the absolute value inequality into two separate inequalities:

• \(-5 \leq 3w + 2\)
• \(3w + 2 \leq 5\)

Then solve each one:- Start with the first inequality: subtract 2 from both sides, and then divide everything by 3 to isolate \(w\), giving \(-\frac{7}{3} \leq w\).- For the second, again subtract 2, divide by 3, resulting in \(w \leq 1\).
Finally, combining these two results, we find that \(w\) is between \(-\frac{7}{3}\) and \(1\). That is \(-\frac{7}{3} \leq w \leq 1\).
Understanding these steps helps in mastering the essential skill of solving inequalities.

Graphing Inequalities

Graphing the solution of an inequality is a visual way to represent all the numbers that satisfy the inequality. It gives us an intuitive understanding of where the solutions lie on the number line.
To graph \(-\frac{7}{3} \leq w \leq 1\), the solution involves marking these critical points on a number line. Because both are 'less than or equal to' inequalities, use closed circles at \(-\frac{7}{3}\) and \(1\) to indicate that these points are included.
Next, shade the region between these points: this shaded area represents all values \(w\) can take in this solution.
By visualizing the graph, students can quickly and clearly see the set of numbers \(w\) might be, solidifying the concept of solutions to inequalities.

Absolute Value Properties

The absolute value of a number represents its distance from zero, disregarding whether it's positive or negative. It's like asking, "How far is this number from zero?" without worrying about direction.
In simpler terms, - is \(|x| \leq a\), which means \(-a \leq x \leq a\). This shows that \(x\) can range from -a to a without crossing boundaries in distance.Absolute value properties are powerful tools for expressing inequalities. They help in translating these absolute expressions into forms we can solve: converting one complex inequality into two simpler equations.
By breaking down \(|3w + 2| \leq 5\) into \(-5 \leq 3w + 2 \leq 5\), students can tackle each part separately, making it much easier to find the solution. Understanding absolute value properties help simplify and effectively approach solving these types of inequalities.