Question

Solve the inequality. Then graph the solution. $$|10-4 x| \leq 2$$

Short answer

The solution to the absolute value inequality \(|10-4x| \leq 2\) is \(2 \leq x \leq 3\). On a number line, this range is illustrated by a segment extending from 2 to 3, inclusive.

Step-by-step solution

Setup Related Equations

Setup the two related equations 10 - 4x = 2 and 10 - 4x = -2. This is because when we have an absolute value inequality like \(|10 - 4x| \leq 2\), it translates to -2 \leq (10 - 4x) \leq 2. So, we solve for x in both cases.

Solve First Equation

Solve the equation \(10 - 4x = 2\). You want to solve for x, therefore, you should first subtract 10 from both sides to get \(-4x = -8\). Then, multiplying each side by -1 to get \(4x = 8\). Lastly, divide by 4 on both sides to isolate the variable x and get \(x = 2\).

Solve Second Equation

Now solve the equation \(10 - 4x = -2\). Here, subtract 10 from both sides to get \(-4x = -12\). Multiplying each side by -1 gives \(4x = 12\). Dividing by 4 on both sides, you get \(x = 3\).

Write Final Solution As Inequality

Write the solution as an inequality. The solutions to the above equations give the bounds for x, thus \(2 \leq x \leq 3\). This means that x is greater than or equal to 2 and less than or equal to 3.

Plot On Number Line

Now that we have the inequality \(2 \leq x \leq 3\), we can plot these values on a number line. The line extends from 2 to 3 with a closed circle at both ends because the inequality includes the values 2 and 3.

Key concepts

Absolute Value

Understanding the concept of absolute value is critical in solving algebraic inequalities. An absolute value is a measure of a number's magnitude regardless of its sign. For example, the absolute value of both \( -3 \) and \( 3 \) is \( 3 \) because distance is always a positive quantity or zero. In equations, we denote the absolute value of a number \( x \) as \( |x| \) which asks the question: 'How far is \( x \) from zero?'

In the context of inequalities such as \( |10 - 4x| \leq 2 \) the absolute value brackets imply two scenarios: the expression inside is either positive and less than or equal to 2, or negative and greater than or equal to -2. We interpret this as a range for \( 10 - 4x \) which leads to two separate inequalities to solve. This dual-condition nature is central to understanding how absolute value operates within inequalities.

Inequality Graphing

Graphing inequalities is a powerful way to visualize solutions to algebraic problems. When dealing with absolute value inequalities, the graph often shows a segment on the number line that represents all the viable solutions. For example, when solving \( |10 - 4x| \leq 2 \), we use a number line to depict the range of values that \( x \) can take.

In our exercise, the values between 2 and 3, inclusive, satisfy the inequality. On a number line, we would draw a line from 2 to 3 with closed circles at each end to indicate these bounds are included, which we refer to as a 'closed interval'. If the inequality did not include the boundary values (such as with '<' or '>'), we would use open circles instead. This visual representation assists in comprehending which values of \( x \) fulfill the given conditions.

Algebraic Equations

An algebraic equation is a statement that two expressions are equal, involving variables and constants. Solving algebraic equations is a foundational skill in algebra, with the goal often being to find the values of variables that make the equation true. In the context of absolute value inequalities, these equations come as a pair to encapsulate both the positive and negative possibilities of the absolute value.

For instance, the absolute value equation \( |10 - 4x| \leq 2 \) breaks down into two separate equations, \( 10 - 4x = 2 \) and \( 10 - 4x = -2 \), which we then solve individually. Through this process, we unravel the range of acceptable solutions for \( x \) and progress towards graphing the inequality. Mastery of manipulating equations by adding, subtracting, multiplying, or dividing both sides is essential to uncover the values of \( x \) that satisfy the equation.

Isolate the Variable

Isolating the variable is a pivotal step in solving algebraic equations and inequalities. It involves manipulating the equation to get the variable by itself on one side of the equation, providing a clear solution. In the case of our absolute value inequality, to isolate \( x \) we perform a series of arithmetic operations.

When we have \( 10 - 4x = 2 \) and \( 10 - 4x = -2 \) from the absolute value inequality \( |10 - 4x| \leq 2 \), the goal is to manipulate these to find the values of \( x \) that render the equations true. This is done by subtracting 10 from both sides and then dividing by -4 to free \( x \) from other terms. It’s important to remember that when multiplying or dividing both sides of an inequality by a negative number, we must flip the inequality sign. However, since we’re dealing with equations first to get the boundary values, we don't need to flip any signs at this stage. After finding the boundary values, we can then craft the final inequality, as was derived in this exercise.