Problem 42 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(\frac{3}{5}(t-2)-\frac{1}{4}(2 t-7) \leq 3\)

Step-by-Step Solution

Answer

t ≤ 24.5; Interval notation: (-∞, 24.5]

Step 1: Distribute the constants

Apply the distributive property to the terms involving the variable. Multiply \(\frac{3}{5}\) by \(t-2\) and \(\frac{1}{4}\) by \(2t-7\). This yields: \[ \frac{3}{5}t - \frac{6}{5} - \frac{1}{4} \times 2t + \frac{1}{4} \times 7 \]

Step 2: Simplify the equations

Combine like terms and simplify: \[ \frac{3}{5}t - \frac{6}{5} - \frac{2}{4}t + \frac{7}{4} \leq 3 \]

Step 3: Common denominator

Find a common denominator for the fractions to combine them more easily: \[ \frac{12}{20}t - \frac{24}{20} - \frac{10}{20}t + \frac{35}{20} \leq 3 \]

Step 4: Combine like terms

Combine the terms with \(t\) and the constant terms: \[ \frac{2}{20}t + \frac{11}{20} \leq 3 \]

Step 5: Simplify the result

Further simplify to: \[ \frac{1}{10}t + \frac{11}{20} \leq 3 \]

Step 6: Clear the fraction

Multiply both sides by 20 to eliminate the fraction: \[ 2t + 11 \leq 60 \]

Step 7: Solve for t

Subtract 11 from both sides: \[ 2t \leq 49 \]Then divide both sides by 2: \[ t \leq 24.5 \]

Step 8: Interval notation

Express the solution in interval notation: \((-\forall, 24.5] \)

Step 9: Graph the solution set

Draw a number line and shade the region to the left of and including 24.5.

Key Concepts

Inequality Solving

Solving inequalities is very similar to solving equations, but there are key differences you must keep in mind. When solving an inequality, you are finding the range of values for the variable that make the inequality true. You use similar steps as when solving an equation: combining like terms, isolating the variable, and using properties of inequalities. However, one important difference is that if you multiply or divide both sides by a negative number, you must reverse the inequality sign. For example, if you have \( -2x > 6 \) and you divide by -2, you get \( x < -3 \). This reversal ensures the inequality remains true.

Distributive Property

The distributive property is a useful algebraic property that lets you multiply a single term by each term within a parenthesis. This is especially handy when dealing with inequalities. For example, in our exercise, we started with \( \frac{3}{5}(t-2)-\frac{1}{4}(2t-7) \). Using the distributive property, we expanded these terms to become: \( \frac{3}{5}t - \frac{6}{5} - \frac{1}{4} \times 2t + \frac{1}{4} \times 7 \). The distributive property helped us to break down and simplify the terms to make it easier to combine like terms later on. Remember, the key idea is to multiply each term inside the parenthesis by the factor outside.

Interval Notation

Using interval notation is a concise way to describe the set of solutions for an inequality. For example, if we have a solution like \( t \leq 24.5 \), we can express this range of values as an interval \( (-\infty, 24.5] \). In interval notation:

• The round bracket ( or ) means the end value is not included (open interval).
• The square bracket [ or ] means the end value is included (closed interval).

So, \( (-\infty, 24.5] \) tells us that the solutions include all values less than or equal to 24.5 but do not include -infinity, since we can't reach it.

Graphing Solutions

Graphing the solution of an inequality involves drawing a number line and shading the portion that represents all the solutions. To graph \( t \leq 24.5 \), you:

• Draw a number line.
• Locate 24.5 on the number line.
• Shade the region to the left of 24.5, which includes all numbers less than 24.5.
• Place a solid dot on 24.5 to indicate that it is included in the solution.

This visual representation helps in understanding the range of possible solutions at a glance and is especially helpful for inequalities involving ranges of values.