Question

Solve the inequality and graph its solution. \(-x+6 \leq 13\)

Short answer

The solution to the inequality is \(x \geq -7\). On the number line, this would be represented with a solid dot on \(-7\) and a line stretching towards positive infinity.

Step-by-step solution

Move the constants to the sides

To isolate \(x\), first we need to move constant terms away from \(x\). Therefore, we subtract \(6\) from both sides of the inequality: \(-x+6-6 \leq 13-6\), which simplifies to \(-x \leq 7\).

Eliminate the negative sign from \(x\)

Next, multiply both sides of the inequality by \(-1\) to get rid of the negative sign on \(x\). However, remember that when you multiply or divide an inequality by a negative number, you must flip the direction of the inequality symbol. Therefore, \(x \geq -7\). This is the solution to the inequality.

Draw the solution on a number line

On a number line, you draw a solid point at \(-7\) to indicate that \(-7\) is included in the solution (due to the \(\geq\) symbol), and then draw a line stretching to the right to the positive infinity to indicate that all numbers greater than \(-7\) are in the solution set.

Key concepts

Graphing Inequalities

Understanding graphing inequalities can be simplified into a process that visually represents the range of possible solutions on a coordinate plane or number line. The process can be summarised with these steps:

• Identify the inequality and solve it algebraically, just as we did with \( -x+6 \leq 13 \).
• Decide whether the graph should use an open circle (which means 'not included') or a closed circle ('included'). In this case, we have \( x \geq -7 \) after flipping the inequality, so we will use a closed circle.
• Shade the region that represents all the possible solutions. If the inequality includes 'greater than', shade to the right; for 'less than', shade to the left. Here, we shade to the right of \( -7 \).

When graphing on a two-dimensional coordinate plane, a similar approach is taken, where the region of possible solutions is identified and shaded accordingly. If the inequality is a linear inequality, the boundary will be a line, and the shading above or below this line represents valid solutions.

Inequality Properties

The properties of inequalities are critical for solving and understanding them correctly. Here are a few key points:

• Addition/Subtraction: You can add or subtract the same number from both sides without changing the direction of the inequality, exemplified by subtracting \( 6 \) from both sides in \( -x+6 \leq 13 \) to derive \( -x \leq 7 \).
• Multiplication/Division: Multiplying or dividing by a positive number does not affect the inequality direction. However, when we multiply or divide by a negative, the inequality must be flipped. This rule was utilized when we multiplied both sides by \( -1 \) to get \( x \geq -7 \).
• Transitive Property: If \( a \leq b \) and \( b \leq c \), then \( a \leq c \) also holds. This might come into play when combining inequalities in more complex problems.
• Division by Zero: Remember, division by zero is undefined, so always check that you're not violating this principle in your calculations.

Inequalities show us the range rather than a specific value. That's why understanding these properties is essential to accurately solving and graphing inequalities.

Number Line Representation

The number line representation gives a straightforward visual of where the solutions to an inequality lie. It's particularly helpful because it breaks down complex algebraic concepts into something that can be visually understood at a glance:

• Every point on the line corresponds to a possible value of \( x \).
• For an inequality like \( x \geq -7 \), which includes the number itself, a closed dot is drawn at \( x=-7 \) on the line.
• The portion of the line representing the solution is shaded or drawn differently to stand out. If the inequality indicates \( x \) is less than a value, the line will be shaded left. If \( x \) is greater, like in our example, it will be shaded to the right.

This method is especially useful when dealing with compound inequalities, where you may need to represent multiple overlapping or non-overlapping intervals.