Problem 57 - Elementary and Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ -6<-3(x-4) \leq 24 $$

Step-by-Step Solution

Answer

The solution set is \([-4, 6)\).

Step 1: Understanding the Inequality

The given compound inequality is \(-6<-3(x-4) \leq 24\). This means we need to find values of \(x\) that satisfy both parts of the inequality simultaneously.

Step 2: Solve the Left Side of the Inequality

Consider the inequality \(-6 < -3(x-4)\). First, distribute the \(-3\) on the right side: \(-6 < -3x + 12\). Then, subtract 12 from both sides to isolate the \(x\) term: \(-18 < -3x\). Divide each side by \(-3\) and remember to flip the inequality sign: \(x < 6\).

Step 3: Solve the Right Side of the Inequality

Now, solve \(-3(x-4) \leq 24\). First, distribute the \(-3\): \(-3x + 12 \leq 24\). Subtract 12 from both sides: \(-3x \leq 12\). Divide each side by \(-3\) and flip the inequality sign: \(x \geq -4\).

Step 4: Combine the Results

The solution must satisfy both \(x < 6\) and \(x \geq -4\). Therefore, the compound inequality solution is \(-4 \leq x < 6\).

Step 5: Graph the Solution

On a number line, \(-4\) is marked with a closed circle (indicating it is included in the solution set), and \(6\) is marked with an open circle (indicating it is not included). Shade the region between \(-4\) and 6.

Step 6: Write the Interval Notation

The interval notation, combining the results of the inequalities, sets the solution as \([-4, 6)\).

Key Concepts

Inequality Solving

Solving an inequality means finding all possible values of a variable that make the inequality true. When you have a compound inequality, you're dealing with two separate inequalities connected by "and" which means both conditions must be satisfied at the same time.

In the example problem,

• The compound inequality is \[-6<-3(x-4) \leq 24\].
• To solve, focus on each part separately.

For the first inequality,

• distribute the \(-3\) inside the parentheses: \[-3(x-4) = -3x + 12\].
• Subtract 12 from both sides: \[-6 - 12 < -3x\].
• It's simplified to \[-18 < -3x\].
• Divide by \(-3\), remembering to **flip** the inequality sign. Thus, \[x < 6\].

For the second inequality,

• distribute \(-3\) again: \[-3x + 12 \leq 24\].
• Subtract 12 from both sides, \[12 - 12 \leq 24 - 12\], to get \[-3x \leq 12\].
• Divide by \(-3\) and **flip** the sign: \[x \geq -4\].

These steps give us: \[-4 \leq x < 6\], meaning \(x\) can satisfy both inequalities at the same time. This solution shows how we methodically handle inequalities and when to flip signs.

Interval Notation

Interval notation simplifies the expression of solutions for compound inequalities by compactly showing the range of numbers that satisfy the inequality. Let's break it down.

Interval notation uses parentheses ___\((, )\)___ and brackets ___\([, ]\)___ to indicate open and closed ends, respectively.

• A bracket \([ \text{ or } ]\)shows that the number next to it is part of the solution, similarly to a "greater/less than or equal to" condition.
• A parenthesis \(( \text{ or } )\)indicates the number next to it is not included in the solution, just like "greater/less than" without the "equal to."

In our solved example, the final solution is \([-4, 6)\).

• The bracket at \(-4\)makes sense because \(x \geq -4\).
• The parenthesis around \(6\)corresponds to \(x < 6\), indicating \(6\) is not a solution.

Interval notation represents solutions clearly and is often used in mathematics, making reading solutions straightforward.

Graphing Solutions

Graphing solutions of inequalities on a number line is a visual way to show which values satisfy the inequality. It's effective in demonstrating continuous solutions, particularly for compound inequalities.

To graph the solution of -4 \(\leq x < 6\):

• Start by drawing a number line, labeling typical numbers like \(-4\) and \(6\).
• Place a closed circle on \(-4\)and an open circle on \(6\).
• The closed circle on \(-4\)indicates that \(-4\)is a part of the solution.
• The open circle at \(6\)shows it isn’t included in the possible values of \(x\).

Once the circles are placed, shade the number line between these two points. This shaded area quickly tells a viewer that all numbers within this range satisfy the inequality. In this case, every number from \(-4\) to just less than \(6\)is a potential value for \(x\). Graphing is helpful for visually confirming solutions and understanding the scope of possible solutions. It's a handy tool for checking work and conveying results.