Problem 49 - Algebra 1: Concepts and Skills

Question

Solve the inequality. Then graph the solution. $$ -14 \leq x+5 \leq 14 $$

Step-by-Step Solution

Answer

The solution to the compound inequality \(-14 \leq x + 5 \leq 14\) is \(-19 \leq x \leq 9\). The solution can be graphically represented as a closed line segment on the number line extending from -19 to 9, inclusive.

Step 1: Simplify the inequality

Start by isolating the variable \(x\) within the compound inequality. Here, we do so by subtracting 5 from all sides of the inequality. We get \(-14 - 5 \leq x + 5 - 5 \leq 14 - 5\) simplifying this further, we get \(-19 \leq x \leq 9\)

Step 2: Interpret the solution

The solution to this compound inequality \(-19 \leq x \leq 9\) means that \(x\) can be any real number such that it is greater than or equal to -19 and less than or equal to 9.

Step 3: Graph the solution

The graph of this solution would be a closed line segment on the number line extending from -19 to 9, inclusive. This is denoted by a filled dot at -19 and 9, and a line segment joining these two points.

Key Concepts

Compound Inequalities

Compound inequalities are mathematical expressions that involve two separate inequalities joined by either "and" or "or". In this case, the inequality \(-14 \leq x+5 \leq 14\) is an example of a compound "and" inequality. This means that both conditions, \(-14 \leq x+5\) **and** \(x+5 \leq 14\), must be satisfied simultaneously. This ensures that the solution set for the value of \(x\) must meet both criteria.

• When solving compound inequalities, it's helpful to work with each part of the inequality separately.
• Simplifying each portion of the inequality individually can pave the way for a clearer understanding and solution.
• Many times, these solutions result in a range of values, expressed as an interval.

For the given inequality, solving each part step-by-step leads us to the final result where \(-19 \leq x \leq 9\). This represents all real numbers between -19 and 9, inclusive on both ends. By understanding the concept of compound inequalities, students can easily interpret similar mathematical expressions and determine their solution sets effectively.

Graphing Inequalities

Graphing inequalities, especially compound ones, involves visual representation of the solution set on a number line. This helps to see which values of \(x\) satisfy the inequality at a glance. For a given inequality such as \(-19 \leq x \leq 9\), we can graph it using our solution. It's key to note:

• The use of 'closed' circles or filled dots indicates that the endpoints are part of the solution.
• For open intervals, omitted dots would indicate the value is not included in the solution set (though this is not applicable in closed intervals like ours).
• Drawing a continuous line between the endpoints shows that all the values in the range are solutions.

Knowing how to graph inequalities allows students to translate abstract mathematical concepts into a concrete visual form, helping in deeper comprehension and verification of solutions.

Number Line Representation

The number line is a powerful visual tool for representing the solutions to inequalities, especially compound ones. It provides a straightforward way to understand the range of possible values that satisfy an inequality.First, mark the endpoints on the line. For our example, place points at -19 and 9.

• First, plot filled circles at -19 and 9.
• Draw a solid line between these points to indicate that all numbers between -19 and 9 are part of the solution.

This representation clearly depicts that all numbers from \(-19\) to \(9\), including the endpoints themselves, are solutions. The use of the number line is essential since it aids students in visualizing mathematical statements and understanding the continuous nature of number sets within inequalities.