Question

Determine whether each value of \(x\) is a solution of the inequality. \(|2 x-3|<15\) (a) \(x=-6\) (b) \(x=0\) (c) \(x=12\) (d) \(x=7\)

Short answer

Out of the four given options, only \(x=7\) is the solution to the inequality \(|2 x-3|<15\), the others are not.

Step-by-step solution

Understanding of absolute value

The term \(|2 x-3|\) means the absolute value of \(2 x-3\), which is the distance of \(2 x-3\) from 0 on the number line. It's always non-negative. Therefore, the inequality \(|2 x-3|<15\) means that the distance from \(2 x-3\) to 0 is less than 15. This can be separated into two inequalities: \(2 x-3<15\) and \(2 x-3>-15\).

Breaking down inequality into two parts

The inequality \(|2 x-3|<15\) can be rewritten as two separate inequalities: \(2x -3 <15\) and \(2x - 3 > -15\). After rearranging these, we get \(x<9\) and \(x>6\). This means that when the value of \(x\) is greater than 6 and less than 9, then \(|2 x -3|<15\). The solution is all \(x\) in the interval (6, 9).

Testing value (a)\(x=-6\)

Substitute \(x=-6\) into the inequality \(|2 x -3 |<15\). The resulting equation becomes \(|-2*(-6)-3|<15\) which simplifies to \(|-12-3|<15\), then \(|-15|<15\), further simplifies to \(15<15\), which is not true. Hence, \(x=-6\) is not a solution to the inequality.

Testing value (b)\(x=0\)

Substitute \(x=0\) into the inequality \(|2 x -3 |<15\). The resulting equation becomes \(|2*0-3|<15\). which simplifies to \(-3<15\), which becomes \(3<15\). The inequality is true. However, 0 is not in our solution set, which is (6, 9). So, \(x=0\) is not a solution to the inequality.

Testing value (c)\(x=12\)

Substitute \(x=12\) into the inequality \(|2 x -3 |<15\). The resulting equation becomes \(|2*12-3|<15\), which simplifies to \(|24-3|<15\), further simplifies to \(21<15\), which is not true. Hence, \(x=12\) is not a solution to the inequality.

Testing value (d)\(x=7\)

Substitute \(x=7\) into the inequality \(|2 x -3 |<15\). The equation becomes \(|2*7-3|<15\). From this, simplification to \(|14-3|<15\), then \(|11|<15\), which will further simplify as \(11<15\). This is true and 7 is in our solution set (6, 9), thus \(x=7\) is indeed a solution to the inequality.

Key concepts

Algebraic Inequalities

When studying algebra, understanding algebraic inequalities is as crucial as dealing with equations. An inequality, much like an equation, relates two expressions. However, unlike equations that show equality, inequalities describe the relationship where one value is larger or smaller than another, using symbols like '<' (less than), '>' (greater than), '(less than or equal to), and '> (greater than or equal to).

For instance, if we have x + 3 > 5, we understand that when three is added to x, it results in a number greater than five. To solve such inequalities, we often perform similar operations as we would in equations to isolate the variable. But attention—when we multiply or divide an inequality by a negative number, we must flip the inequality sign. This preserves the true relationship between the sides. Knowing these rules and applying them consistently is fundamental to navigating the complexities of algebraic inequalities.

Absolute Value

The notion of absolute value can sometimes be puzzling. Simply put, it represents the distance of a number from zero on a number line, without regard to direction. What this means is that the absolute value of any number is always non-negative, as distance cannot be negative. When we see an expression within bars like \( |x| \), it's asking for the absolute value of x.

For example, the absolute value of both \( -4 \) and \( 4 \) is \( 4 \) because both are four units away from zero. In the context of inequalities, dealing with absolute values can lead to two scenarios—either the expression inside the absolute value is positive or it is negative. The absolute value bars are sometimes referred to as 'distance bars', symbolizing the concept of distance in a real-world sense. Demystifying this is vital as it's a stepping stone to working through more complex problems involving absolute value inequalities.

Inequality Solution Testing

Solving inequalities is an important skill, but testing solutions is an equally important part of the process. Testing the solutions ensures that we've not only found a potential range of answers but also that these answers are valid for the given inequality.

When dealing with absolute value inequalities, finding the solution set typically involves considering two cases. Once the potential solutions are found, we plug these values back into the original inequality to confirm their validity. Depending on whether the inequality holds true, the tested values are either accepted or rejected as solutions. It's a verification step that legitimizes our work and builds confidence in our problem-solving abilities—cementing the notion that through methodical testing, we reaffirm the correctness of our solutions.