Problem 52 - Beginning and Intermediate Algebra

Question

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$|4 k+9| \leq 5$$

Step-by-Step Solution

Answer

The solution set for the absolute value inequality \(|4k + 9| \leq 5\) is the interval \(\left[-\dfrac{7}{2}, -1\right]\).

Step 1: Identify the inequality

We are given the absolute value inequality: \(|4k + 9| \leq 5\)

Step 2: Rewrite the absolute value inequality as two separate linear inequalities

Since the absolute value of an expression is less than or equal to a positive number, we can rewrite the inequality as two separate inequalities with opposite signs: \(4k + 9 \leq 5\) and \(-(4k + 9) \leq 5\) Now, we simplify the second inequality to get: \(-4k - 9 \leq 5\)

Step 3: Solve both linear inequalities

First, we'll solve the inequality \(4k + 9 \leq 5\): 1. Subtract 9 from both sides: \(4k \leq -4\) 2. Divide both sides by 4: \(k \leq -1\) Next, we'll solve the inequality \(-4k - 9 \leq 5\): 1. Add 9 to both sides: \(-4k \leq 14\) 2. Divide both sides by -4 (remember to reverse the inequality sign when dividing by a negative number): \(k \geq -\dfrac{7}{2}\)

Step 4: Write the solution in interval notation

We found the solutions \(k \leq -1\) and \(k \geq -\dfrac{7}{2}\). To write this in interval notation, we write the simultaneous solution of both inequalities as an interval: \[\left[-\dfrac{7}{2}, -1\right]\] So the solution set for the inequality \(|4k + 9| \leq 5\) is the interval \(\left[-\dfrac{7}{2}, -1\right]\).

Key Concepts

Absolute Value Equations

Absolute value equations involve absolute value expressions. The absolute value, represented by vertical bars \(|x|\), refers to the distance of a number from zero on the number line, without considering its sign. This means \(|x| = x\) if \(x\) is positive or zero, and \(|x| = -x\) if \(x\) is negative. Absolute value equations can be rewritten as two separate equations. For example, if you have an equation like \(|x - 3| = 7\), it can be rewritten as two separate equations: \(x - 3 = 7\) and \(x - 3 = -7\). Solving these will give you the solutions \(x = 10\) and \(x = -4\). Understanding this helps solve absolute value inequalities, as they too need to be expressed in a form that eliminates the absolute value before solving.

Linear Inequalities

Linear inequalities are similar to linear equations but involve inequality signs such as \(<, \leq, >,\) and \(\geq\) instead of an equal sign. They express a range of possible values rather than a specific solution. For example, in the solution step for the absolute value inequality \(|4k + 9| \leq 5\), we decomposed it into two linear inequalities: \(4k + 9 \leq 5\) and \(-4k - 9 \leq 5\). Each of these can be solved through operations similar to those used for equations, like addition, subtraction, multiplication, or division. It's essential to remember that when multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

Set Notation

Set notation is a way to describe a set of numbers that satisfy certain conditions. When solving equations, set notation expresses the solution as a set of values that make the equation true. For example, if you solved an equation and found solutions \(x = 2\) or \(x = 5\), you could write this in set notation as \(\{2, 5\}\). Set notation is concise and helps to clearly convey the solution set for equations, distinguishing them from inequalities, which use interval notation to express their solutions.

Interval Notation

Interval notation is used to represent the set of all real numbers between two endpoints. It is particularly useful for expressing solutions to inequalities. In interval notation, square brackets \([\text{and}\, ]\) are used to include an endpoint, while parentheses \((\text{and}\, )\) are used to exclude it. For instance, an inequality solution such as \(k \leq -1\) and \(k \geq -\dfrac{7}{2}\) can be expressed in interval notation as \([-\dfrac{7}{2}, -1]\). This means that \(k\) can be any value from \(-\dfrac{7}{2}\) to \(-1\), inclusive of both endpoints. Interval notation provides a clear and efficient means to express the range of values that make an inequality true.