Problem 40 - Algebra and Trigonometry

Question

In all exercises, other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a mumber line. In Exercises \(27-50,\) solve each linear inequality. $$ 5(3-x) \leq 3 x-1 $$

Step-by-Step Solution

Answer

The solution to the inequality \(5(3-x) \leq 3x - 1\) is \(x \leq 2\). In interval notation, this is expressed as \([- \infty, 2]\).

Step 1: Distribute

First, distribute '5' into the parenthesis: \(5(3-x) = 15 - 5x\). Therefore, the inequality becomes: \(15 - 5x \leq 3x - 1\).

Step 2: Rearrange the inequality

Next, combine like terms by moving all x-terms to one side and constants to the other side of the inequality. Therefore, add \(5x\) to both sides and add 1 to both sides to get: \(8x \leq 16\).

Step 3: Solve for x

Isolate the variable by dividing both sides of the inequality by 8. The solution then become: \(x \leq 2\).

Step 4: Interval notation & Number line

The solution to the inequality, \(x \leq 2\), in interval notation is \([- \infty, 2]\) and includes 2. On the number line, this is shown by drawing a solid point at 2 (indicating it is included in the solution set) and shading to the left (indicating any number less than 2 is in the solution set).

Key Concepts

Interval Notation

Interval notation is a concise way of expressing the set of all numbers that satisfy a given inequality. It is a means of communication that clearly and unambiguously specifies which numbers are included in a set. In the worked example concerning the inequality \(5(3-x) \textrm{leq} 3x - 1\), the solution after the algebraic manipulations resulted in \(x \leq 2\). This is represented in interval notation as \([-\infty, 2]\). The square bracket, \([\), indicates that the number 2 is part of the solution, and the inequality includes all numbers up to and including 2. Conversely, the \(-\infty\) symbol represents all numbers less than 2 extending indefinitely, and it is always accompanied by a parenthesis \((\), because infinity is not a number we can reach or include in a set.

It's essential to recognize that using interval notation contributes to clarity and precision when expressing solutions, especially when dealing with real numbers. With practice, interpreting and writing in interval notation becomes a streamlined process that aids in understanding the range of solutions to inequalities and other mathematical conditions.

Inequality Graphing

Graphing inequalities on a number line helps visualize the set of all possible solutions. It's a powerful tool in understanding how inequalities behave. In our example, the solution \(x \leq 2\) is graphed on a number line. To do this, a solid circle is placed on the number 2, because the number is included in the solution set (as reasoned from the inclusive inequality \(\leq\)). Then, a shaded line is extended to the left of 2 to represent all numbers lesser than 2 that also satisfy the inequality. The solid circle contrasts with an open circle, which would be used if the number at that point were not included in the solution set (for example, if the inequality were \(x < 2\)).

Such graphical representation aids students in better grasping the concept of intervals and makes comparing and contrasting different inequalities more intuitive. When inequalities are sketched on a number line, the direction of the shade (left for less than, right for greater than) and the type of circle (solid for \'included\', open for \'not included\') must always be considered for an accurate representation.

Algebraic Manipulation

Algebraic manipulation is fundamental in solving inequalities. It’s the process of rearranging the equation to isolate the variable of interest, in this case, \(x\). For the given problem, the initial step is distributing the multiplier across the terms within the parentheses, which streamlines the inequality to a more manageable form. From \(5(3-x) \leq 3x - 1\), we reached \(15 - 5x \leq 3x - 1\) through distribution.

The next stage involves consolidating like terms. By moving all terms containing \(x\) to one side and the numerical constants to the other, we simplify the equation further, leading to \(8x \leq 16\). The final act of manipulation requires dividing by the coefficient of \(x\), which here is 8, to solve for the variable. This clear-cut approach ensures that students can systematically address inequalities without confusion. Mastery of algebraic manipulation not only aids in solving inequalities but also equips students with versatile problem-solving tools applicable across various mathematical challenges.