Question

Sketch the graph of the inequality. $$-3 x \geq 15$$

Short answer

The solution to the inequality \(-3x \geq 15\) is \(x \leq -5\). The x-values that satisfy the inequality are all real numbers less than or equal to -5.

Step-by-step solution

Dividing both sides of the inequality by -3

To isolate x, divide both sides of the inequality by -3. Beware that dividing or multiplying an inequality by a negative number flips the inequality sign. Therefore, the inequality changes from \(-3x \geq 15\) to \(x \leq -5\)

Plotting the solution on a number line

The solution to this inequality is a semi-infinite interval that includes all real numbers less than or equal to -5. Therefore, on the number line, make a solid dot at -5 to show that this point is included in the solution set and then draw a line extending to the left from this point to show the range of numbers that x can take on.

Key concepts

Solving Inequalities

Solving inequalities is similar to solving equations, but with one critical difference—how we treat the inequality sign when multiplying or dividing by a negative number. For example, when we have an inequality like \(-3x \geq 15\), our objective is to isolate the variable to one side to determine its possible values. To do this, we divide both sides of the inequality by -3. This is where we encounter the unique rule for inequalities: when you divide both sides by a negative number, the direction of the inequality sign must be reversed. This step changes our original inequality to \(x \leq -5\), providing us with the set of all values for 'x' that will make the inequality true.

It's imperative to remember that whenever you perform an operation involving a negative number on both sides of an inequality, you must flip the inequality sign. It ensures that the relational aspect of the inequality remains accurate. This can be a common pitfall for students, so it's crucial to keep this rule in mind to successfully solve inequalities.

Number Line Representation

Representing solutions to inequalities on a number line is an effective way to visualize the set of all possible values that satisfy the inequality. For the inequality \(x \leq -5\), let's understand how to graph this solution. Firstly, we draw a horizontal line which represents the number line. On this line, we locate the point corresponding to -5. As the inequality includes the number -5 (indicated by the 'less than or equal to' sign \(\leq\)), we place a solid dot on -5 to indicate that it is part of the solution set.

Following this, because our variable 'x' can take any value less than -5, we draw a line extending to the left from our solid dot, towards the negative direction, and often we draw an arrow at the end to show that it continues indefinitely. This pictorial representation effectively shows all the numbers that are solutions to the inequality in an easy-to-understand format.

Inequality Sign Reversal

Understanding when and why the inequality sign reverses is crucial in solving inequalities. The reversal arises due to the properties of inequalities themselves. Unlike equalities, where the balance is maintained regardless of whether a positive or negative number is used, with inequalities, the relationship between the quantities changes based on the direction of the imbalance.

For instance, if we multiply or divide both sides of an inequality by a negative number, the sense of the inequality changes. Why does this happen? Consider the inequality \(-3x \geq 15\). If '-3x' were positive, increasing 'x' would increase '-3x'. However, because '-3x' is negative, increasing 'x' actually makes '-3x' smaller. Thus, reversing the inequality sign when dealing with negative coefficients keeps the integrity of the relationship intact. It ensures that if we test any value for 'x' that is less than or equal to -5, the original inequality \(-3x \geq 15\) will hold true, illustrating the importance of this rule.