Problem 66 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$8(z-2)<4(z+1)$$

Step-by-Step Solution

Answer

The solution to the inequality is \( z < 5 \).

Step 1: Distribute

Distribute the multipliers to the numbers in brackets. This gives: \( 8z - 16 < 4z + 4 \).

Step 2: Simplify the inequality by collecting like terms

Subtract \( 4z \) from both sides to keep the inequality balanced. Also, add 16 to both sides. This results in: \( 4z < 20 \).

Step 3: Isolate z

Divide both sides of the inequality by 4 to isolate z, resulting in: \( z < 5 \).

Step 4: Graph the solution

On a number line, place an open circle at 5 (an open circle indicates that 5 is not included in the solution). Then, draw a line to the left of 5 to indicate all numbers less than 5 are part of the solution.

Key Concepts

Algebraic Inequalities

Understanding algebraic inequalities is essential for solving problems where you're working with ranges instead of exact values. An inequality is like an equation, except instead of saying two expressions are equal, it states that one expression is greater than or less than another. In the given problem, we see an inequality that starts out with brackets: \(8(z - 2) < 4(z + 1)\). The initial step is to distribute the multipliers, which simplifies to \(8z - 16 < 4z + 4\).

With these inequalities, balancing acts are crucial. Simplifying further by collecting like terms and maintaining the balance on both sides gives us \(4z < 20\), which when divided by 4 leaves us with \(z < 5\). This tells us that z can be any number less than 5, offering a range of values as a solution.

Graphing Inequalities

Graphing is a powerful tool for visualizing inequalities and their solutions. Unlike typical equations, where we only plot a point or a line, inequalities often require shading a region. For instance, in our example inequality \(z < 5\), we represent this on a number line.

To graph this inequality, you start by placing an open circle at 5, which signifies that 5 is not included in the set of solutions—that's why we don't fill in the circle. After marking this critical value, we need to show that every number smaller than 5 is a solution. We do this by drawing a line or arrow to the left, towards the smaller numbers, often shading this region to indicate that these are all acceptable values for z.

Inequality Notation

Inequality notation is the system we use to write down inequalities precisely and concisely. Consisting of symbols like \(<\), \(>\), \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to), this notation helps us understand at a glance the relationship between values in an inequality.

It's important to use the correct symbols. For example, \(z < 5\) denotes that z is any number less than 5, but doesn't include 5 itself. If 5 were a valid solution, we would use \(z \leq 5\) instead. Paying attention to whether the symbol has a line under it (\(\leq\) or \(\geq\)) tells us if the range include the number we're comparing to, vital for correctly interpreting and solving these mathematical expressions.