Problem 103

Question

Use the graph of $y-|4-x|$ to solve each inequality. $|4-x|<5$

Step-by-Step Solution

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Answer

The solution to the inequality $|4-x|<5$ is $(-1, 9)$. This means that any value between $-1$ and $9$ will satisfy the inequality.

1Step 1: Interpret the absolute value inequality

Rewrite the absolute value inequality $|4-x|<5$ as a distance from zero. This means the distance between $4-x$ and $0$ is less than $5$. It can be interpreted as $4-x$ is between $-5$ and $5$.

2Step 2: Formulate the inequality without the absolute values

The above understanding can be translated into two inequalities. This gives us $-5<4-x<5$.

3Step 3: Isolate x in the inequality

To find the solution, isolate x in the inequality by performing the same operation on each part of the inequality. Subtracting 4 from each part, we get $-5-4<-x<5-4$. Simplifying this gives $-9<-x<1$. However, notice that there is a minus sign before 'x'. So to eliminate the negative sign we multiply entire inequality by $-1$. However, remember that when we multiply an inequality by a negative number, the inequality signs will flip, resulting in $9>x>-1$. This is the same as \(-1

4Step 4: Write the solution in interval notation

The last step is to write the solution in interval notation. This inequality represents all x-values that lie between $-1$ and $9$ but excluding $-1$ and $9$. So, the solution in interval notation is $(-1, 9)$.

Key Concepts

Absolute ValueInequality NotationInterval Notation

Absolute Value

Understanding the concept of absolute value is crucial in solving inequalities that contain them. The absolute value of a number is the distance of that number from zero on the number line, regardless of direction. For instance, both 3 and -3 have an absolute value of 3 because they are both exactly three units away from zero on the number line.

When solving the inequality like the one given, $ |4-x|<5 $ , we interpret the expression inside the absolute value, $ 4-x $ , as having a distance from zero that’s less than 5. This will apply in both directions on the number line, thus, we split the inequality into two parts: one for the positive direction and one for the negative. Hence, we maintain the 'distance' but remove the absolute value symbols, leading to two separate inequalities that need to be solved.

Inequality Notation

Solving inequalities often involves rewriting them in different forms to isolate the variable of interest. Inequality notation is a shorthand way of representing the relationship between variables and numeric values. The symbols '<' and '>' are used to show that something is less than or greater than something else, respectively.

In the given exercise, the inequality $ -5<4-x<5 $ uses inequality notation to show that the value of $ 4-x $ must be more than -5 but less than 5. Through manipulation, like subtraction and multiplication, we aim to solve for 'x'. Importantly, when we multiply or divide both sides of an inequality by a negative value, the direction of the inequality symbols must be reversed to preserve the inequalities’ true meaning. This rule is essential to correctly solve absolute value inequalities.

Interval Notation

After solving the inequalities, we represent the solution using interval notation, which is a compact way of describing sets of numbers. This notation is particularly useful when expressing the solution set of an inequality.

For example, the solution to the inequality \( -1

Problem 102

Problem 103

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