Problem 87 - Algebra 1: Concepts and Skills

Question

Solve the inequality. Then graph and check the solution. $$ |5+2 x| \leq 7 $$

Step-by-Step Solution

Answer

The solution to the inequality |5+2x| ≤ 7 is \(-6 \leq x \leq 1\). These values can be visualized on a number line, with the portion between -6 and 1 shaded to reflect the solution set.

Step 1: Rewrite the inequality without absolute value

The first step is to express the inequality without the absolute value. From property of absolute values, we know that for any real numbers \(a\) and \(b\), if \(|a| \leq b\), this means \(-b \leq a \leq b\). Hence, for our inequality \(|5+2x| \leq 7\), we can rewrite it as \(-7 \leq 5+2x \leq 7\).

Step 2: Split the inequality and solve

First, let's take the inequality \(-7 \leq 5+2x\). If we subtract 5 from both sides, we get \(-12 \leq 2x\). Dividing both sides by 2, we get \(-6 \leq x\). \n\nSimilarly, take the other inequality \(5+2x \leq 7\). Subtract 5 from both sides to get \(2x \leq 2\). Then, divide both sides by 2 to get \(x \leq 1\). Therefore, the solution to our inequality is \(-6 \leq x \leq 1\)

Step 3: Graph and check the solution

First, graph the solution on a number line by shading the region between -6 and 1 inclusive. Then, use a couple of test points to check the validity of the solution. For example, test points \(x = -6, 0, 1\). All of these values yield an absolute value less than or equal to 7 when plugged back into the original inequality, therefore confirming the solution is correct.

Key Concepts

Solving Inequalities

Solving inequalities is a core concept in algebra, dealing with finding the range of values that satisfy a given inequality condition. Let's delve deeper into our exercise to truly grasp this concept. If you're given an inequality like \(|5 + 2x| \leq 7\), the absolute value inequality tells us we need to consider both positive and negative scenarios because of the nature of absolute values. This leads to creating two separate inequalities: \(-7 \leq 5 + 2x\) and \(5 + 2x \leq 7\).

• The first inequality \(-7 \leq 5 + 2x\) simplifies to \(-6 \leq x\) after subtracting 5 from both sides and then dividing by 2.
• The second inequality \(5 + 2x \leq 7\) simplifies to \(x \leq 1\) using the same operations.

By solving these inequalities, we find that the overall solution is \(-6 \leq x \leq 1\). This means any number between -6 and 1, inclusive, will satisfy the original absolute value inequality condition.

Graphing Inequalities

Graphing inequalities helps us get a visual understanding of the solutions and see which values satisfy our inequalities. For the solution \(-6 \leq x \leq 1\), we graph this on a number line, which makes understanding easier. Here is how you can do it:

• Start by drawing a horizontal line. Indicate numbers that are key to the inequality: in this case, mark -6 and 1.
• Now, shade the region on the number line between -6 and 1. This shading shows all the possible solutions for \(x\) that satisfy the inequality.
• Both -6 and 1 are included, so use closed circles or dots at these endpoints. Closed circles indicate that the value itself is part of the solution.

Tracing these steps on a number line gives a clear picture of where solutions lie, aiding both in understanding and checking our work.

Compound Inequalities

Compound inequalities are essentially two simple inequalities combined into a single statement, where both conditions must be true simultaneously. In our example, \(-6 \leq x \leq 1\) is a compound inequality, meaning it combines \(x \geq -6\) and \(x \leq 1\).

• To solve compound inequalities, you address each part individually until you achieve a single solution set, as we did above.
• Compound inequalities are common where the solution is a range, rather than a singular value, making effectively solving these vital in many mathematical contexts.

Understanding and solving compound inequalities allow us to capture the full set of solutions where both conditions are satisfiable, ensuring our solutions are thorough and accurate.