Problem 57 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Write the solution in interval notation. $$|7-4 x| \leq 11$$

Step-by-Step Solution

Answer

The solution is \([-1, 4.5]\).

Step 1: Define the Absolute Value Inequality

The given inequality is \(|7 - 4x| \leq 11\). This means the expression inside the absolute value can range between -11 and 11: \(-11 \leq 7 - 4x \leq 11\).

Step 2: Solve the Right Inequality

Start with the inequality \(7 - 4x \leq 11\):\[ 7 - 4x \leq 11. \]Subtract 7 from both sides to isolate the \(-4x\) term:\[ -4x \leq 4. \]Finally, divide both sides by -4, remembering to reverse the inequality sign, to solve for \(x\): \[ x \geq -1. \]

Step 3: Solve the Left Inequality

Now solve \(-11 \leq 7 - 4x\):\[ -11 \leq 7 - 4x. \]Subtract 7 from both sides:\[ -18 \leq -4x. \]Divide by -4, and reverse the inequality sign:\[ x \leq 4.5. \]

Step 4: Combine the Solutions

From both solved inequalities, we have \(x \geq -1\) and \(x \leq 4.5\). Combining these, the solution is \(-1 \leq x \leq 4.5\).

Step 5: Write the Solution in Interval Notation

The solution in interval notation, expressing the set of all \(x\) values that satisfy the inequality, is \([-1, 4.5]\).

Key Concepts

Absolute Value Inequality

When we talk about absolute value inequalities, we are exploring expressions that use the absolute value sign, such as \(|a| \leq b\).
The absolute value symbol \(| \, |\) represents the distance of a number from zero on the number line, which is always positive or zero. In the inequality \(|7 - 4x| \leq 11\), the expression inside the absolute value can swing between -11 and 11. This means we have two scenarios to solve:

• One side for the positive range: \(7 - 4x \leq 11\)
• And one for the negative range: \(-11 \leq 7 - 4x\)

Each part requires solving a different inequality to find the range of values that satisfy the original expression.

Interval Notation

Interval notation is a way to express a range of numbers that satisfy an inequality. It’s a concise way to denote all the possible values of \(x\) that make an inequality true.
In interval notation, you use brackets or parentheses:

• A square bracket \[ \] means the number is included in the interval.
• A parenthesis \( \) means the number is not included.

For instance, the solution \([-1, 4.5]\) tells us that \(x\) can be any number starting at \(-1\) and ending at \(4.5\), including both endpoint values.
This helps clearly define the set of all numbers that satisfy the given inequality.

Solving Inequalities

Solving inequalities involves finding the values of the variable that make the inequality true.
The critical steps are similar to solving equations, but with some extra rules:

• If you multiply or divide both sides by a negative number, you must reverse the inequality sign.

In solving our exercise, we dealt with two inequalities:

• First, solving \(7 - 4x \leq 11\) involved removing constants and isolating \(x\).
• Next, solving \(-11 \leq 7 - 4x\) required similar steps but keeping track of sign changes when dividing by a negative.

These procedures ensure that we accurately capture the full set of solutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent a mathematical relationship or rule. \(7 - 4x\) is an example of such an expression.
Understanding how to manipulate and solve algebraic expressions is key to working with inequalities.The expression in our inequality had variables that we needed to isolate, giving us insights into the possible values of \(x\). By properly simplifying and rearranging terms:

• We moved constants to one side.
• We dealt with the coefficient of the variable.

These skills help not only in inequalities but also in broader algebraic problems, laying the foundation for more advanced math.