Problem 26 - Algebra 1: Concepts and Skills

Question

Solve the inequality. $$ -x+9 \geq 14 $$

Step-by-Step Solution

Answer

x <= -5

Step 1: Re-arrange the inequality

One goal is to isolate the variable x. To do this, begin by subtracting 9 from both sides of the inequality. As 9 is on both sides of the inequality, it cancels out on the left side and we are left with -x >= 14 - 9.

Step 2: Simplify the right hand side

The next step is to perform the subtraction on the right side. This results in -x >= 5.

Step 3: Eliminate the negative sign

The variable x should be positive. In order to turn -x into x, we multiply both sides of the inequality by -1. This flips the inequality, because when we multiply both sides of an inequality by a negative number, the direction of the inequality changes. This results in x <= -5.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a cornerstone of solving equations and inequalities. This involves rearranging and simplifying expressions to isolate variables and simplify equations.
In the context of inequalities, our primary goal is to solve for the variable of interest. In the given problem, the inequality \(-x + 9 \geq 14\) involves manipulating the terms to isolate \(x\).

• Start by eliminating constant terms from one side of the inequality, which involves subtracting or adding the same number on both sides.
• Next, you must simplify any expression, like performing subtraction on constants, to make the inequality more straightforward.

These manipulations make it easier to see the bounds or constraints on the variable, helping to find the solution.

Inequalities

Inequalities are similar to equations but instead of an equal sign, they use inequality symbols such as \(<, >, \leq,\) and \(\geq\). These symbols indicate the relationship between expressions rather than exact equality.
Understanding inequalities is crucial because:

• They allow for multiple solutions, as long as the relationship holds true. For example, \(x \leq -5\) means any number less than or equal to \(-5\) satisfies the inequality.
• The rules for manipulating inequalities are similar to those for equations, with an important exception when multiplying or dividing by a negative number, which flips the inequality sign.

In algebra, inequalities help in understanding and describing ranges of possible solutions rather than pinpointing one exact solution.

Negative Numbers

Working with negative numbers involves an understanding of how operations affect them. Whenever you multiply or divide both sides of an inequality by a negative number, the direction of the inequality changes.
This is a unique property of inequalities, and it is crucial to ensuring the inequality remains correct.

• Negative signs in equations or inequalities can be tackled by flipping the sign when needed, as seen when converting \(-x\) to \(x\).
• It's essential to carefully perform operations, particularly multiplication and division, since improper handling may result in incorrect solutions.

Keeping these rules in mind will help simplify expressions involving negative numbers, allowing for more accurate solutions to inequalities.