Problem 64 - Introductory Algebra for College Students

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. \(-3 x+14<5\)

Step-by-Step Solution

Answer

The solution is \(x > 3\). The graph on the number line would start at 3 (not included), and include all numbers greater than 3.

Step 1: Solve the inequality

To begin with, the inequality is \(-3x + 14 < 5\). The goal is to isolate \(x\). Start by subtracting 14 from both sides, which yields \(-3x < -9\).

Step 2: Further solve for \(x\)

Next, divide both sides by -3. Remember to flip the inequality sign when dividing by a negative number. This gives \(x > 3\).

Step 3: Drawing the number line

Draw a number line, place a solid circle at 3 to represent that 3 is not part of the solution set, and draw an arrow to the right to show that all solutions greater than 3 are included.

Key Concepts

Addition Property of Inequality

The addition property of inequality is essential for solving inequalities as it helps to modify them while keeping the inequality's balance. If you have an inequality like \( a < b \), you can add the same number to both sides of the inequality to achieve \( a + c < b + c \) without changing the truth of the original inequality. This property is handy when trying to isolate a variable. In the exercise \( -3x + 14 < 5 \) we used this property by subtracting 14 from both sides to start isolating x, leading to \( -3x < -9 \).

The key point to remember when applying the addition property is that the inequality sign will not change direction unless you're adding negative numbers. Always maintain the balance of the inequality, just as you would do with an equation.

Multiplication Property of Inequality

Similarly to addition, the multiplication property of inequality also keeps the truth of an inequality intact, providing it's used correctly. With any inequality, you can multiply or divide both sides by the same positive number, and the direction of the inequality will remain the same. For instance, \( a < b \) implies \( ac < bc \) when \( c \) is positive. However, if \( c \) is negative, the inequality flips: \( ac > bc \). That's exactly what happens in our example when we go from \( -3x < -9 \) to \( x > 3 \), as we divide by -3, which is negative.

It's crucial to remember this rule since forgetting to reverse the inequality when dividing by a negative number is a common mistake that leads to the wrong solution set.

Number Line Graphing

Graphing on a number line gives a visual representation of the solution set of an inequality. To graph \( x > 3 \) on a number line, you start by drawing a horizontal line, which represents all possible real numbers. Place a mark on the line at the number 3. Since 3 is not included in the solution set \( x > 3 \) (it's not \( x \geq 3 \) ), we use an open circle to symbolize this exclusion. Then, draw a ray extending rightwards from the open circle because the variable \( x \) represents all numbers greater than 3.

This visual aid helps in understanding which numbers satisfy the inequality and can be a great tool when dealing with more complex inequalities or when comparing multiple inequalities.

Isolate Variable

To isolate the variable means to manipulate the inequality in such a way that the variable of interest is left on one side of the inequality sign, and everything else is on the other side. This process often involves a combination of using the addition and multiplication properties of inequality. In the given problem, we first isolate \( x \) by eliminating the constant term on the same side as \( x \) and then by dealing with the coefficient multiplying by \( x \). The goal is to have \( x \) alone on one side, as seen in the final result \( x > 3 \).

Mastery of isolating the variable is key for solving inequalities as it turns an inequality with a variable term into an easier to interpret and a more usable form, which can then be graphed or otherwise evaluated.