Question

SOLVING INEQUALITIES Solve the inequality. $$-y-3 x \leq 6$$

Short answer

The solution to the inequality is \(y \leq -6 - 3x\).

Step-by-step solution

Re-arrange the inequality

First, the inequality \(-y-3x \leq 6\) can be re-arranged into \(6 \geq -y - 3x\).

Isolate \(y\)

Next, isolate the variable \(y\) by moving \(-3x\) from the right side of the inequality to the left side. This gives \(6 + 3x \geq -y\).

Switch the direction

In order to switch the side of \(y\), we multiply by -1, which also involves flipping the inequality sign. Therefore, the final solved inequality will be \(y \leq -6 - 3x\).

Key concepts

Linear Inequality

When dealing with a linear inequality, such as the given problem \( -y - 3x \leq 6 \), we're working with an algebraic expression that involves a '<' or '>' sign instead of the '=' sign found in linear equations. A linear inequality looks to find all the possible values that can make the inequality true. The difference here is that instead of looking for a specific value for \( y \), we are looking for a range of values. Unlike equations, where \( x \), and \( y \) correspond to a single point on a graph, in inequalities, \( x \) and \( y \) represent a region.

Linear inequalities are a fundamental concept because they illustrate how to deal with constraints and conditions within various fields such as economics, engineering, and physics. When solving linear inequalities, it is important to remember that multiplying or dividing both sides by a negative number reverses the inequality sign. This characteristic is a key difference from linear equations and is crucial to the correct solution of these problems.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions. It is a key skill in solving both equations and inequalities. To manipulate an algebraic expression correctly, one must adhere to the fundamental rules of algebra, which ensures that the inequality remains balanced. In the example given, \( -y - 3x \leq 6 \), algebraic manipulation involves several steps to isolate the variable \( y \).

Firstly, the expression can be adjusted to place the constant term on the left to get \( 6 \geq -y - 3x \). This reordering does not affect the inequality because the relationship between the terms remains the same. After that, moving terms involving \( x \) to the opposite side results in \( 6 + 3x \geq -y\).

Attention to Detail

Finally, we multiply through by -1, which is a critical point. The multiplication by -1 necessitates reversing the inequality to maintain the true relationship between both sides, leading to \( y \leq -6 - 3x \). When teaching algebraic manipulation, emphasizing these subtleties enhances student understanding and avoids common mistakes.

Inequality Graphing

Graphing an inequality gives a visual representation of all the potential solutions to the inequality. Unlike linear equation graphing, which produces a line, graphing an inequality results in a shaded region. For the inequality \( y \leq -6 - 3x \), the graphing process begins with the related equation \( y = -6 - 3x \) to plot the boundary line. This particular line would slope downward due to the negative coefficient of \( x \) and it would intersect the \( y \) axis at \( -6 \). The inequality \( y \leq -6 - 3x \) indicates we are interested in the area below this line, which includes all points that satisfy the inequality.

Shading & Boundary Lines

When graphing, it's crucial to mark the line as either solid or dashed. A solid line is used when the inequality includes an equal sign (\leq or \geq), signifying that points on the line are included in the solution set. A dashed line is used when the inequality is strict (less than or greater than without the equal sign), indicating the points on the line aren’t part of the solution. Then, by selecting a test point not on the line (typically \(0,0\) if it’s not on the line) and substituting into the original inequality, one can determine which side of the boundary line to shade, completing the solution process. The shaded region represents all possible points \( (x,y) \) that satisfy the inequality.