Question

Solve the inequality. Then match its solution with its graph. (GRAPH CAN'T COPY) \(x+4 \leq 6\)

Short answer

The solution for x in the inequality \(x+4 \leq 6\) is \(x \leq 2\). On a graph, it would be shown as a line extending to the left from 2, with a closed dot at 2.

Step-by-step solution

Analyze the inequality

The given inequality is \(x+4 \leq 6\). The goal is to solve this inequality for the variable \(x\). This means we need to isolate \(x\) on one side of the inequality.

Solve the inequality

To isolate \(x\), subtract 4 from both sides of the inequality. So, \[x + 4 - 4 \leq 6 - 4\]which simplifies to \[x \leq 2\]

Interpretation of the Solution

The solution \(x \leq 2\) means that any value of \(x\) less than or equal to 2 will satisfy the inequality. Although the graph can't be displayed here, on a number line, it would be shown as a line extending to the left from 2, with a closed dot at 2 to indicate that 2 itself is included in the solution.

Key concepts

Inequality Solutions

Solving inequalities is a fundamental skill in algebra that allows you to find the range of values that make an inequality true. Unlike equations, which have exact solutions, inequalities describe a set of possible solutions. For example, when facing an inequality such as \(x + 4 \leq 6\), the ultimate goal is to determine which values of \(x\) satisfy the condition.

To solve this particular inequality, it's essential to perform operations that preserve the inequality relationship, such as subtracting the same number from both sides. By doing so, the inequality \(x + 4 - 4 \leq 6 - 4\) simplifies to \(x \leq 2\). The solution, \(x \leq 2\), implies that any number less than or equal to 2 makes the original inequality true. It's important to remember that when multiplying or dividing both sides of an inequality by a negative number, the inequality sign should be flipped to maintain the proper relationship.

Algebraic Inequalities

Algebraic inequalities, such as the one given \(x + 4 \leq 6\), are expressions that involve a comparison between values. The inequality symbols \(<\), \(>\), \(\leq\), and \(\geq\) signify 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively. Understanding these symbols is crucial for interpreting and solving inequalities.

When solving algebraic inequalities, follow the same principles as you would for equations, with careful attention to the direction of the inequality. Adding or subtracting the same quantity from each side, or multiplying or dividing by positive quantities, will not change the inequality's direction. However, multiplying or dividing by negative quantities requires reversing the inequality symbol. It's also important to present the solution set in the simplest possible way so anyone can easily understand it.

Number Line Graphing

Graphing the solution of an inequality on a number line helps visualize the range of possible values that satisfy the inequality. For instance, the solution to \(x \leq 2\) means every number less than or equal to 2 is a solution. On a number line, this is represented by a solid dot at 2 and a shaded line extending to the left, indicating all the numbers less than 2 are included.

It's important to recognize the difference between a closed dot and an open dot in such graphs. A closed dot at 2 means that 2 is a part of the solution set, aligning with the \(\leq\) symbol. If the symbol were instead \(<\), an open dot would be used, illustrating that 2 is not included. Graphing on a number line serves as a powerful visual aid to support the understanding of inequality solutions, making it easier to observe the spread of values that make the inequality true.