Question

Solve the inequality and graph the solution. $$-x+7>13$$

Short answer

The solution to the inequality \(-x+7>13\) is \(x<-6\). The graph is a number line with a point at -6 and a line extending to the left of -6, with an open circle at -6.

Step-by-step solution

Rearrange the Inequality

Rearrange the inequality by subtracting 7 from both sides to isolate the term including x. So, \(-x+7-7>13-7\) simplifies to \(-x>6\).

Solve for x

To solve for x, divide both sides by -1. However remember that when dividing or multiplying an inequality by a negative number, the direction of the inequality changes. Thus, \(-x/-1<6/-1\) gives \(x<-6\).

Graph the Solution

The solution of the inequality is \(x<-6\). On the number line, plot a point at -6 and draw a line going left from -6 to represent all numbers less than -6. Because the inequality is '<', instead of '<=', the point at -6 should be an open circle, meaning -6 is not included in the solution.

Key concepts

Algebraic Inequalities

When solving algebraic inequalities, you are finding all the values of a variable that make the inequality true. Inequalities are mathematical expressions involving the symbols <, >, \( \le \) (less than or equal to), and \( \ge \) (greater than or equal to). These symbols are used to compare two values or expressions. Unlike equations, inequalities do not have just one solution; they describe a range of possible solutions.

For instance, in the inequality \( -x + 7 > 13 \), the goal is to isolate \( x \) on one side to find the values that satisfy the inequality. We carry out similar steps as we would in solving an equation—adding, subtracting, multiplying, or dividing both sides by the same number. However, a unique rule for inequalities is that when you multiply or divide both sides by a negative number, the inequality sign flips. This is an essential step to remember, as this ensures the inequality remains true. After simplifying \( -x + 7 > 13 \) and applying these rules, we arrive at \( x < -6 \) as the solution set.

Graphing Inequalities

Graphing is a powerful tool that visually represents the solution to an inequality. The steps taken to graph an inequality can give students a clear picture of the set of all possible solutions. In the example \( -x+7>13 \), after solving the inequality algebraically, we found that \( x<-6 \) is the solution. To graph it, you start with a number line and identify the critical point, which is -6 in this case.

Since our inequality is a strict inequality (\( x < -6 \)), meaning it does not include -6 itself, we use an open circle to represent this on the number line. An open circle indicates that the point is not included in the solution set. Next, we draw a ray or line extending in the direction that represents all the numbers that are less than -6, which would be to the left of the number line since the number line increases in value from left to right. This visual representation shows at a glance all the values that satisfy the inequality.

Inequality Number Line

The inequality number line is a simple yet effective way to represent the solutions to an inequality. It consists of a horizontal line marked with numbers like a ruler, where each point corresponds to a real number. To use a number line for inequalities, identify whether the inequality is 'strict' (\( < \) or \( > \)) or 'non-strict' (\( \le \) or \( \ge \)).

• For a strict inequality, one that does not include the number itself, we draw an open circle.
• For a non-strict inequality, which includes the number, we use a closed circle.

Then, we draw an arrow or a line starting at the circle that covers all possible solutions. For example \( x<-6 \) on a number line, you would place an open circle at -6 and draw a line extending to the left. This indicates all numbers less than -6 are solutions. The direction of the arrow shows whether to include larger or smaller values than the number at the circle.