Question

Solve the inequality. Then sketch a graph of the solution on a number line. $$|9-2 x|+3<4 \quad $$

Short answer

The solution to the inequality \(|9-2 x|+3<4\) is \(4 < x < 5\). On a number line, this is represented as an open interval between 4 and 5.

Step-by-step solution

Simplify the Inequality

First thing, as the inequality involves an absolute value, it's crucial to get rid of the terms not linked directly to the absolute expression. The inequality is \(|9 - 2x| + 3 < 4\). To simplify it, subtract 3 from both sides. This leads to the simple absolute value inequality: \(|9 - 2x| < 1\)

Solve the Absolute Value Inequality

An absolute value inequality \(|a| < b\) is equivalent to the combined inequalities \(-b < a < b\). Applying this to \(|9 - 2x| < 1\), gives \(-1 < 9 - 2x < 1\).

Solve for x

Now, deal with the compound inequality by isolating x on both sides. To get x, subtract 9 from each part of the inequality, giving \(-1 -9 < -2x < 1 -9\) which simplifies to \(-10 < -2x < -8\). After that, divide all parts by -2, but remember that dividing by a negative number reverses the inequality symbols, which gives \(5 > x > 4\). This inequality can also be written as \(4 < x < 5\).

Sketch the Solution on a Number Line

Finally, draw a number line and represent the solution. The solution is the open interval \( (4, 5) \), hence it includes all real numbers between 4 and 5, but not 4 and 5 itself. Draw an open circle over 4 and 5 which indicates that these numbers are not included in the solution. And draw a line that connects these two circles to represent the solution set, being all numbers in between.

Key concepts

Understanding Absolute Value

When we refer to the 'absolute value' of a number, we're talking about its distance from zero on a number line, without considering its direction. Think of it as the 'numerical distance' and not its actual position. So in both directions, whether you go left or right from zero, the absolute value remains the same because distance is always positive or zero.

More formally, the absolute value of a number 'a', denoted as \( |a| \), is defined as:

• \( |a| = a \) if \( a \geq 0 \)
• \( |a| = -a \) if \( a < 0 \)

In essence, if you have a negative number, you flip it to positive by removing the negative sign, and if it's positive or zero, it stays as it is.

Deciphering Number Line Graphs

Number line graphs give us a visual way to represent and solve inequalities and absolute value equations. Imagine the number line as a horizontal 'ruler' where each point stands for a possible solution to an equation or inequality.

To graph an interval, such as \( (4, 5) \) which represents all numbers between 4 and 5 but not including 4 and 5 themselves, we use:

• Open circles to show that 4 and 5 are not part of the solution set.
• A solid line connecting these circles to indicate that every number in between is included in the solution.

This method gives a quick and clear representation of all possible answers that satisfy an inequality.

Navigating Compound Inequalities

Compound inequalities are essentially two (or more) inequalities joined together by 'and' or 'or'. That's like having two separate conditions that our solution needs to meet at the same time.

When solving compound inequalities such as \( -10 < -2x < -8 \), you treat each part of the inequality separately first, and then combine the solutions. An 'and' compound inequality means the solution must satisfy both conditions, resulting in the intersection of the two solutions, which often looks like a segment or a point on the number line.

Solving Algebraic Inequalities

Algebraic inequalities are a lot like algebraic equations but with one critical difference - they show a relationship where one side is not equal to, but greater or less than the other side. For example, \( x + 2 > 5 \) means 'x plus 2 is greater than 5'.

To solve them, you perform similar steps as you would with an equation - like adding, subtracting, multiplying, or dividing both sides - but you must remember a key rule: if you multiply or divide by a negative number, you must reverse the inequality sign. This prevents you from getting misleading solutions and ensures the number line representation matches the true solution set.