Problem 80 - College Algebra

Question

Solve absolute value inequality. \(5|2 x+1|-3 \geq 9\)

Step-by-Step Solution

Answer

The solution to the inequality \(5|2 x+1|-3 \geq 9\) will be x values that satisfy either of the two resulting inequalities from breaking down the absolute value.

Step 1: Isolate the Absolute Value

Add 3 to both sides to isolate the term that includes the absolute value. This results in \(5|2x+1| \geq 12 \)

Step 2: Divide both sides by the coefficient of the absolute value

Divide both sides by 5 to further isolate the absolute value. This results in \(|2x+1| \geq \frac{12}{5} \)

Step 3: Break down the Absolute Value

Break down the absolute value into two separate inequalities. \(2x+1 \geq \frac{12}{5}\) and \(2x+1 \leq -\frac{12}{5}\).

Step 4: Solve the Inequalities

Subtract 1 from both sides of both inequalities. Result is: \(2x \geq \frac{12}{5}-1\) and \(2x \leq -\frac{12}{5}-1\), solve each to get solutions for x.

Step 5: Finalize the Solutions

After solving the inequalities, you will have two ranges for the possible values of x. Bear in mind that they represent different ranges and therefore are joined by the word 'or' to include all possible solutions.

Key Concepts

Solving Inequalities

Inequalities describe the relative size or order of two values. Solving inequalities involves finding all possible values that make the inequality true. Unlike equations, which have exact solutions, inequalities often have a range of solutions. For example, let's consider the inequality from our exercise, where we aim to solve for the variable 'x'.

When solving inequalities, we perform similar operations as with equations, such as adding, subtracting, multiplying, or dividing both sides by the same number. However, there's a critical difference: when we multiply or divide both sides of an inequality by a negative number, we must flip the direction of the inequality sign. This ensures the inequality remains true. Think of inequality solution sets as segments on the number line that show what values 'x' can take.

Absolute Value Equations

Absolute value measures the distance of a number from 0 on the number line, irrespective of direction. An absolute value equation is an equation that contains an absolute value expression. Solving absolute value equations is somewhat unique because the absolute value of a number can be both positive and negative.

For example, if we have an equation like \(|x| = a\), the solutions are \(x = a\) and \(x = -a\), because both \(a\) and \(-a\) are at the same distance from 0 on the number line. An important aspect of absolute value equations is breaking them down into two separate cases, one where the term inside the absolute value is positive and the other where it's negative, which we see in Step 3 of our exercise solution.

Algebraic Inequalities

Algebraic inequalities are equations where the two sides are not necessarily equal but rather have a relation expressed by inequality signs such as '<', '>', '\(\leq\)', or '\(\geq\)'. Inequalities can be more complex when they involve absolute values, variables on both sides, or higher powers.

When working with algebraic inequalities, it's crucial to keep the inequality balanced, just like with equations, unless we multiply or divide by a negative number. In the context of our exercise, the inequality we have is an absolute value inequality. The challenge here is to consider both directions of the absolute value when breaking it down into algebraic inequalities without the absolute value, as demonstrated in Step 3 of our solution. Remember that the inequalities give us a range of solutions, which we represent on a number line to visually display all possible solutions.

Isolate Absolute Value

Isolating the absolute value is a foundational step in solving absolute value inequalities. To isolate the absolute value means to get the absolute value expression by itself on one side of the inequality sign. This is similar to isolating the variable in a regular equation and requires performing arithmetic operations to both sides until the absolute value term is on its own.

In our exercise, we start by adding 3 to both sides, followed by dividing both sides by 5. These steps eliminate any coefficients or constants attached to the absolute value (as seen in Steps 1 and 2), thus setting up the stage to break the problem into two simpler algebraic inequalities. The isolated absolute value gives a clear starting point to consider the two cases where the term inside the absolute value can be positive or negative.