Problem 11 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$6 q+4 \leq 28$$

Step-by-Step Solution

Answer

The solution is \(q \leq 4\). On the number line, shade all values to the left of 4, including 4 itself.

Step 1: Isolate the Variable Term

Start by isolating the variable term, which is \(6q\), on one side of the inequality. To do this, subtract 4 from both sides of the inequality. \[ 6q + 4 - 4 \leq 28 - 4 \] This shows that \[ 6q \leq 24 \]

Step 2: Solve for the Variable

To solve for \(q\), divide both sides of the inequality by 6:\[ \frac{6q}{6} \leq \frac{24}{6} \] This simplifies to:\[ q \leq 4 \] The inequality \(q \leq 4\) is the solution.

Step 3: Check the Solution

Choose a number less than or equal to 4, such as 3, to check if the solution satisfies the original inequality.Substitute \(q = 3\) back into the original inequality:\[ 6(3) + 4 \leq 28 \] This results in:\[ 18 + 4 \leq 28 \] which simplifies to\:\[ 22 \leq 28 \] The statement is true, verifying that the solution is correct.

Step 4: Graph the Solution on a Number Line

To represent \(q \leq 4\) on a number line, draw a line and shade the region extending to the left from 4. Include a closed circle at 4 to indicate that it is part of the solution set. This shows that any number less than or equal to 4 is a solution to the inequality.

Key Concepts

Number Line Graphing

In order to clearly visualize the solutions for an inequality like \(q \leq 4\), using a number line is an excellent approach. Imagine a horizontal line that represents all possible numbers. The number 4 is marked on this line, and since \(q\) can be 4 or less, you draw a circle around the number 4.
Since the inequality is "less than or equal to," this circle is closed, showing that 4 is included in the solution set.
You then draw a line extending from 4 towards the left, shading in the region to show all numbers that satisfy \(q \leq 4\).

• Closed circle at 4 means 4 is included.
• Shade to the left because all numbers less than 4 are part of the solution.
• Use arrows to indicate the direction for an infinite set of numbers.

This graph visually communicates every number that makes the inequality true, aiding in understanding which values work.

Variable Isolation

To solve an inequality like \(6q + 4 \leq 28\), isolating the variable is essential. You start by getting \(q\) alone on one side. This tells you the range of values \(q\) can take.
First, eliminate constant terms on the side of the inequality containing the variable. Here, subtract 4 from both sides: \[ 6q + 4 - 4 \leq 28 - 4 \] This simplifies to: \[ 6q \leq 24 \] Now, divide both sides by the coefficient of \(q\), which is 6: \[ \frac{6q}{6} \leq \frac{24}{6} \] Finally, simplify to find: \[ q \leq 4 \]

• Subtract to eliminate constant terms.
• Divide to isolate \(q\).
• Simplify to find the variable's maximum value.

This step-by-step isolation process is crucial in solving inequalities effectively.

Checking Solutions

Checking your solutions ensures accuracy in solving inequalities. Once you have obtained a solution, it's important to verify it by substituting values back into the original inequality.
For this case, try \(q = 3\), which should satisfy \(q \leq 4\). Substitute into the original equation: \[ 6(3) + 4 \leq 28 \] This simplifies to: \[ 18 + 4 \leq 28 \] \[ 22 \leq 28 \] The statement holds true, confirming that your solution is indeed correct.

• Select a test value that is part of the solution set.
• Substitute back into the original inequality.
• Verify that the inequality holds true.

By checking, you confirm that all steps were executed correctly and the solution is valid. This practice is essential for confidence and precision in solving inequalities.