Question

Solve the inequality and graph its solution. \(m+7 \geq-10 \quad\)

Short answer

The solution to the inequality is \(m \geq -17\), and its graphical representation includes a closed dot at -17 with the line shaded to the right.

Step-by-step solution

Isolate the variable

To isolate \(m\), subtract 7 from both sides of the inequality. This yields \(m+7-7 \geq -10-7\), simplified to \(m \geq -17\).

Graph the solution

Draw a number line and mark the point -17 on it. Since the inequality includes an equal sign, the point -17 is included in the solution set. Represent this with a closed dot or bracket. The inequality \(m \geq -17\) means that \(m\) is any number greater than or equal to -17, hence shade the number line to the right of the point -17.

Key concepts

Inequality Graphing

Graphing an inequality like the given problem, which is to solve and graph the solution for the inequality \(m+7 \geq -10\), is a pivotal skill in understanding how inequalities work.

Graphing translates the algebraic representation into a visual one, making it easier to understand and interpret the solution set. To graph an inequality, one must first solve the algebraic inequality for the variable in question, which is \(m\) in this instance. After isolating \(m\), the solving process shows that \(m \geq -17\).

As per the solution, you visualize this inequality on a number line. You start by plotting a point at -17. Because the inequality includes an equal sign (\(\geq\)), meaning \(-17\) is a valid solution, a solid dot or closed bracket should be used to denote where the equality portion includes the endpoint.

Then, since \(m\) is greater than or equal to -17, you shade the number line to the right of -17, to indicate all numbers that are greater than -17 are included in the solution set. This graphical representation is a direct and efficient way to portray inequalities, enhancing understanding and providing a quick reference for the set of possible solutions.

Isolate the Variable

The process of isolating the variable is a fundamental step when solving algebraic inequalities and equations alike. It involves manipulating the algebraic expression in order to have the unknown variable on one side of the inequality or equation, and all numerical values on the other side.

This tactical move, crucial to understanding and solving algebraic problems, is neatly exhibited in the exercise’s step by step solution. First, you subtract 7 from both sides of the initial inequality to cancel out the +7 added to \(m\), leading to the simplified inequality \(m \geq -17\).

Why Isolating the Variable Matters

Isolating the variable allows you to clarify what range of values the variable can include. It simplifies the inequality to a point where interpretation becomes straightforward, helping to ensure you fully understand which values of the variable make the inequality true, and subsequently aiding in the creation of the graph that represents the inequality solution.

Number Line Representation

A number line representation is one of the simplest yet most effective ways to express the solution of an inequality. It provides a linear visualization of all possible solutions.

After solving the inequality algebraically and isolating the variable, the next step is to represent this solution graphically. The number line is drawn horizontally, with marks representing numbers in an ascending order from left to right. In the context of the given inequality, the point -17 is of particular interest.

Using Closed Dots and Shading

You use a closed dot at -17 to indicate that -17 itself is a part of the solution. Shading is then applied to the number line starting at -17 and extending rightward, indicating the direction of increasing numbers. This shaded area on the number line comprises all the numbers that satisfy the inequality \(m \geq -17\), signifying that any number on the line that is to the right of -17—including -17 itself—is a solution to the inequality. The clear depiction on the number line ensures no confusion resides with regards to which numbers are included in the solution set, enhancing comprehension for students tackling inequalities.