Question

Solve the inequality. Then sketch a graph of the solution on a number line. $$|5+x|+4 \leq 11$$

Short answer

The solution to the inequality is \( -12 \leq x \leq 2 \). In the number line, both -12 and 2 are included in the solution set, so the interval is closed on both ends.

Step-by-step solution

Isolate Absolute Value

Start by isolating the absolute value. This is done by subtracting 4 from both sides of the inequality: \(|5 + x| \leq 7\).

Address Both Absolute Value Conditions

Next, set up two inequalities to account for the absolute value, one for when the inside is positive and one for when it's negative. This gives us \(5 + x \leq 7\) and \(-(5 + x) \leq 7\).

Solving the First Inequality

Solving the first inequality for x gives us \(x \leq 2\).

Solving the Second Inequality

For the second inequality, distribute the negative sign across \((5 + x)\) to get \(-5 - x \leq 7\). Solve this for x to get \(x \geq -12\).

Graphing the Solution on a Number Line

Plot both the solutions on a number line. As \(x\) lies between -12 and 2, the solution set should be shown as a line segment starting at -12 and ending at 2, with both endpoints included.

Key concepts

Understanding Absolute Value Equations

When we're dealing with absolute value equations, it's essential to remember that the absolute value of a number refers to its distance from 0 on a number line, without considering direction. This distance is always non-negative. For example, both 3 and -3 have an absolute value of 3.

An absolute value equation is an equation that contains an absolute value expression. Solving these equations often involves considering two separate cases because the expression inside the absolute value can be either positive or negative, but the output after applying the absolute value is always positive or zero.

Let's look at our example: \[|5 + x| + 4 \leq 11\]. Removing the constant term on the left by subtracting 4 from both sides leaves us with the absolute value isolated: \[|5 + x| \leq 7\]. This represents all x values whose distance from -5 is less than or equal to 7.

Graphing Inequalities

Visualizing Solutions on a Graph

Inequality graphing allows us to visually understand the range of solutions to an inequality. An inequality like \(x \leq 2\) would be represented on a graph with a solid dot at 2 and a shaded line extending to the left, indicating that all numbers less than or equal to 2 are solutions.

Similarly, for \(x \geq -12\), you would place a solid dot at -12 and shade the line to the right. The intersection of these two shaded regions shows the numbers that are solutions to both inequalities, providing us with the solution to the original absolute value inequality.

Number Line Representation

Visualizing on a Number Line

Using a number line to represent inequalities can be an efficient way to grasp the range of possible solutions. Absolute value inequalities can sometimes involve a segment of the number line where the endpoints represent the maximum and minimum solutions. For our example, after simplifying the inequalities to \(x \leq 2\) and \(x \geq -12\), we indicate the solutions by marking the endpoints at -12 and 2 and shading the entire segment between them. Solid circles at -12 and 2 signify that these endpoints are included in the solution set, which means any x-value within that segment, including -12 and 2, satisfies the original inequality.

Algebraic Manipulation

Importance of Algebra Skills

Strong algebraic manipulation skills are vital for solving absolute value inequalities. The ability to rearrange equations and inequations is at the heart of cracking these problems. As you observed, distributing the negative and then adding or subtracting terms to isolate the variable, x, were crucial steps. Mastering these skills allows students to convert complex expressions into simpler ones that are easier to solve or graph, just like turning \( - (5 + x) \leq 7 \) into its equivalent, \( x \geq -12 \).