Problem 71 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -2 \leq 12-2(x-3) \leq 20 $$

Step-by-Step Solution

Answer

The solution set is \([-1, 10]\).

Step 1: Simplify the Expression

The given inequality is \[-2 \leq 12 - 2(x - 3) \leq 20.\]Start by simplifying the expression inside the inequality by distributing the \(-2\) across \((x - 3)\). This gives:\[-2 \leq 12 - 2x + 6 \leq 20.\]Combine the constants within the inequality:\[-2 \leq 18 - 2x \leq 20.\]

Step 2: Isolate the Variable

Next, we isolate \(x\). Begin by subtracting 18 from all parts of the inequality:\[-2 - 18 \leq 18 - 2x - 18 \leq 20 - 18,\]which simplifies to:\[-20 \leq -2x \leq 2.\]

Step 3: Solve for x

To solve for \(x\), divide all parts of the inequality by \(-2\). When dividing by a negative number, reverse the inequality sign:\[\frac{-20}{-2} \geq x \geq \frac{2}{-2}.\]Simplifying this gives:\[10 \geq x \geq -1.\]

Step 4: Write the Interval Notation

The solution set in interval notation is written from the lowest value to the highest value for \(x\):\[[-1, 10].\]

Step 5: Graph the Solution Set

Draw a number line. Place closed circles or brackets on \(-1\) and \(10\), as these are inclusive in the solution. Shade the region between \(-1\) and \(10\) to represent all values of \(x\) that satisfy the inequality.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of numbers, particularly representing the set of solutions to an inequality on the real number line. It uses brackets and parentheses to describe the numbers in the set.

• Brackets "[" or "]" are used to indicate that an endpoint is included in the set (closed interval).
• Parentheses "(" or ")" indicate that an endpoint is not included (open interval).

Understanding how to express ranges correctly is crucial for interpreting inequalities. In our problem, after solving the inequality, we found the solution for the variable in terms of a range. This range, [-1, 10], means:

• The solutions for the variable start from -1 and extend to 10, inclusive of both endpoints.
• This is why the interval notation uses closed brackets, denoting that both -1 and 10 are part of the solution set.

Graphing Inequalities

Graphing inequalities involves shading portions of the number line to indicate where the inequality holds true. It's a visual representation that complements the algebraic solution. Here are the key steps for graphing:

• Identify the endpoints. These are the solutions from the interval notation, here -1 and 10.
• Determine whether these endpoints should be marked with closed circles (indicating inclusive) or open circles (indicating exclusive). In this exercise, we use closed circles because our interval notation is closed.
• Shade the region along the number line between these endpoints. This shaded region represents all the possible "x" values satisfying the inequality.

Graphing helps in understanding the solution by making it clear which values of "x" meet the inequality constraints. It is particularly useful in checking for potential mistakes in algebraic manipulation by visually verifying the range of solutions.

Solving Inequalities Step by Step

Solving inequalities involves a series of methodical algebraic steps, much like solving equalities or equations. However, inequalities have additional rules, especially when dealing with negative numbers.

Simplifying Expressions

Start with breaking down complex parts of the inequality using basic operations such as distribution and combination of like terms. For example, in the original inequality, distributing -2 and combining constants simplified the expression.

Isolating the Variable

The next goal is to isolate the variable (like "x") on one side of the inequality. This often involves adding, subtracting, multiplying, or dividing through the inequality.

Remember the Inequality Direction

One of the crucial aspects is remembering to reverse the inequality sign when multiplying or dividing by a negative number. This rule is vital and often a point of error.

Write the Solution and Graph It

Finally, express the solution in interval notation and graph it for visual confirmation. These steps ensure that the solution is both accurate and easy to understand.