Problem 16 - Elementary and Intermediate Algebra

Question

Write the inequality \(10>|16 x-3|\) in an equivalent form with the absolute value expression on the left side.

Step-by-Step Solution

Answer

The inequality with absolute value on the left side is \(|16x - 3| < 10\).

Step 1: Understand the Absolute Value Inequality

To solve the inequality \(10 > |16x - 3|\), we first recognize it as an absolute value inequality, which states that the expression inside the absolute value must be within \(-10\) and \(10\). This means that the inequality can be split into two separate inequalities: \(-10 < 16x - 3 < 10\).

Step 2: Solve the First Inequality

Start with the first part of the inequality: \(-10 < 16x - 3\). Add \(3\) to both sides to begin isolating \(x\):\[-10 + 3 < 16x\]\[-7 < 16x\]Next, divide by \(16\) to solve for \(x\):\[\frac{-7}{16} < x\]

Step 3: Solve the Second Inequality

Now solve the second part of the inequality: \(16x - 3 < 10\). Add \(3\) to both sides:\[16x < 10 + 3\]\[16x < 13\]Then divide by \(16\) to solve for \(x\):\[x < \frac{13}{16}\]

Step 4: Combine the Results

Combine both solutions from Steps 2 and 3 to get the final compound inequality:\[-\frac{7}{16} < x < \frac{13}{16}\]

Step 5: Rewrite the Inequality with the Absolute Value on the Left Side

The original inequality \(10 > |16x - 3|\) is equivalent to having the absolute value expression on the left side, like so:\[|16x - 3| < 10\]

Key Concepts

Solving Inequalities

When tackling inequalities, the approach varies slightly from solving equations. The reason is that inequalities express a range of solutions rather than a specific value. This range tells us about all the values that satisfy our inequality. In our exercise, we have the inequality: \(10 > |16x - 3|\). This inequality is solved by considering the expression inside the absolute value to be within certain bounds. By understanding these bounds, we can split the absolute value inequality into two simpler inequalities. This forms the key to unlocking the solution to inequalities involving absolute values.

Absolute Value Expressions

Absolute value expressions involve the absolute value of a number, which is essentially its distance from zero on the number line, ignoring its sign. So, \(|a| = a\) if \(a \geq 0\) and \(|a| = -a\) if \(a < 0\). In our exercise's inequality, \(10 > |16x - 3|\), the goal is to determine values of \(x\) such that the expression inside the absolute value, \(16x - 3\), stays within a range. This means identifying the range as \(-10 < 16x - 3 < 10\). Solving these gives us a set of values for \(x\) within the parameters defined by the absolute value. Remember, this transformation into two separate inequalities paints the picture of the values between which the original expression must fall.

Compound Inequalities

Compound inequalities involve more than one inequality joined together, usually by the logical word 'and' or 'or'. In this exercise, after breaking down the absolute value inequality, we obtain two simpler inequalities: \(-10 < 16x - 3\) and \(16x - 3 < 10\). These form what's known as a compound inequality, which we solve by resolving each part. To isolate \(x\), we handle each inequality independently but combine their solutions at the end. This results in the range \(-\frac{7}{16} < x < \frac{13}{16}\). This tells us that \(x\) is sandwiched between these two values, forming a clear and understandable range where our original inequality holds true. Understanding compound inequalities is crucial as they appear often when working with absolute values.