Problem 22 - Algebra 2 and Trigonometry

Question

In \(13-22,\) solve each equation or inequality. Each solution is an integer. $$ -3-2 x \geq 12+x $$

Step-by-Step Solution

Answer

The solution is \(x \leq -5\).

Step 1: Move All Terms Involving 'x' to One Side

Start by subtracting \(x\) from both sides of the inequality to bring all terms containing \(x\) to one side. This will give us:\[-3 - 2x - x \geq 12\] which simplifies to \[-3 - 3x \geq 12\].

Step 2: Move Constant Terms to the Other Side

Add 3 to both sides of the inequality to isolate the \(-3x\) term:\[-3 - 3x + 3 \geq 12 + 3\] which simplifies to \[-3x \geq 15\].

Step 3: Solve for 'x'

Finally, divide both sides of the inequality by \(-3\) to solve for \(x\). It is important to remember that when you divide or multiply an inequality by a negative number, the inequality sign flips:\[x \leq \frac{15}{-3}\] which simplifies to \[x \leq -5\].

Key Concepts

Integer Solutions

When solving inequalities, sometimes the solutions are required to be integers. An integer is any whole number, whether positive, negative, or zero. Whole numbers don't include fractions or decimals. For example, -5, 0, 6 are all integers.

In the given problem, after rearranging and simplifying terms, we ended up with the inequality \(x \leq -5\). This inequality doesn't only have -5 as a solution, but all integers less than or equal to -5, such as -6, -7, etc. These are considered integer solutions.

Remember, every time you solve an inequality, ensure you check if integer solutions are required. Sometimes inequalities yield results that aren't integers. But if the exercise specifies that solutions must be integers, only consider whole numbers within the range indicated by the inequality.

Inequality Manipulation

Inequality manipulation involves performing operations that alter the form of an inequality while preserving its truth. Similar to equations, but with an important caveat - the direction of the inequality sign. In the given example, the manipulation involved isolating terms and rearranging the inequality. Here’s a breakdown:

• Combine Like Terms: First, gather all terms containing the variable on one side and the constants on the other. This ensures a smoother transition into solving the inequality.
• Maintain Inequality Direction: When dividing or multiplying by a negative number, flip the inequality sign. This maintains the logical consistency of the inequality.
• Check Solutions: At every step, ensure that the solution set remains true to the inequality. This solidifies the validity of your solution.

Practicing these manipulations can make solving inequalities more intuitive. Inequality manipulation is a powerful tool in algebra, allowing the reshaping of expressions to reveal the solution set.

Variable Isolation

Isolating variables in an inequality is central to finding a solution. It involves rearranging the inequality so that the variable of interest is alone on one side of the inequality sign.

Here's how it was achieved in the example problem:

• First, subtract the variable term from each side to consolidate it. This concentrates the variable on one side, leaving constants on the other.
• Next, simplification continued by adding or subtracting constants. This eliminated excess numbers, moving the inequality progressively toward a simpler form.
• Finally, dividing by the coefficient of the variable permits isolation. Remember: when dividing or multiplying by a negative coefficient, flip the inequality direction. This step finalizes the isolation process and completes the solving method.

Variable isolation isn't exclusive to inequalities and is a universal skill in solving equations. Mastering this means being able to find solutions swiftly and efficiently.