Problem 28
Question
Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ 5 x-3 \geq 2 \text { and } 6 \geq 4 x-3 $$
Step-by-Step Solution
Verified Answer
The solution is \([1, \frac{9}{4}].\)
1Step 1: Solve the First Inequality
The first inequality is \(5x - 3 \geq 2\). To isolate \(x\), add 3 to both sides: \[5x - 3 + 3 \geq 2 + 3\] which simplifies to \[5x \geq 5\]. Next, divide both sides by 5: \[x \geq 1\].
2Step 2: Solve the Second Inequality
The second inequality is \(6 \geq 4x - 3\). First, add 3 to both sides: \[6 + 3 \geq 4x - 3 + 3\] which simplifies to \[9 \geq 4x\]. Now, divide both sides by 4: \[\frac{9}{4} \geq x\] or equivalently, \(x \leq \frac{9}{4}\).
3Step 3: Find the Overlapping Solution Set
The solution to the compound inequality requires both conditions to be true simultaneously. So, combine \(x \geq 1\) and \(x \leq \frac{9}{4}\). The overlapping solution is \(1 \leq x \leq \frac{9}{4}\).
4Step 4: Graph the Solution Set
On a number line, plot a closed circle at 1 and a closed circle at \(\frac{9}{4}\). Shade the region between 1 and \(\frac{9}{4}\) to denote all values in this interval are part of the solution.
5Step 5: Express in Interval Notation
The interval notation for the solution set is \([1, \frac{9}{4}]\). This notation indicates that 1 and \(\frac{9}{4}\) are included in the solution set.
Key Concepts
Solving InequalitiesInterval NotationGraph Solution Sets
Solving Inequalities
In mathematics, inequalities involve the use of signs like \( >, <, \geq, \leq \) to show one side is larger or smaller than the other. When solving inequalities, the goal is to find all values of a variable that make the inequality true.
For example, consider the inequality \(5x - 3 \geq 2\). To solve it, we aim to isolate \(x\). Start by adding 3 to both sides to cancel out the negative 3, which gives us \(5x \geq 5\).
Now, divide both sides by 5 to isolate \(x\), resulting in \(x \geq 1\). Follow similar steps for the second inequality \(6 \geq 4x - 3\): add 3 to both sides, resulting in \(9 \geq 4x\), then divide by 4 to find \(x \leq \frac{9}{4}\).
When working with compound inequalities, finding a common solution set or the overlap of the solutions is key. Here, both conditions \(x \geq 1\) and \(x \leq \frac{9}{4}\) must be satisfied at the same time. Hence, the solution is the interval \(1 \leq x \leq \frac{9}{4}\).
For example, consider the inequality \(5x - 3 \geq 2\). To solve it, we aim to isolate \(x\). Start by adding 3 to both sides to cancel out the negative 3, which gives us \(5x \geq 5\).
Now, divide both sides by 5 to isolate \(x\), resulting in \(x \geq 1\). Follow similar steps for the second inequality \(6 \geq 4x - 3\): add 3 to both sides, resulting in \(9 \geq 4x\), then divide by 4 to find \(x \leq \frac{9}{4}\).
When working with compound inequalities, finding a common solution set or the overlap of the solutions is key. Here, both conditions \(x \geq 1\) and \(x \leq \frac{9}{4}\) must be satisfied at the same time. Hence, the solution is the interval \(1 \leq x \leq \frac{9}{4}\).
Interval Notation
Interval notation is a concise way to express a range of values, typically used with inequalities. It involves brackets and parentheses to indicate whether endpoints are included or excluded.
In our exercise, the solution set \(1 \leq x \leq \frac{9}{4}\) can be expressed using interval notation as \([1, \frac{9}{4}]\).
The square brackets \([\) mean that the endpoints 1 and \(\frac{9}{4}\) are included in the solution set. If an endpoint was not included, a parenthesis \(()\) would be used instead. For instance, \((1, \frac{9}{4}]\) would indicate 1 is not part of the solution.
Interval notation streamlines the expression and can easily be interpreted on a number line.
In our exercise, the solution set \(1 \leq x \leq \frac{9}{4}\) can be expressed using interval notation as \([1, \frac{9}{4}]\).
The square brackets \([\) mean that the endpoints 1 and \(\frac{9}{4}\) are included in the solution set. If an endpoint was not included, a parenthesis \(()\) would be used instead. For instance, \((1, \frac{9}{4}]\) would indicate 1 is not part of the solution.
Interval notation streamlines the expression and can easily be interpreted on a number line.
Graph Solution Sets
Graphing solution sets visually represents the range of solutions for an inequality on a number line. It’s an intuitive method to showcase which values satisfy the given conditions.
To graph \(1 \leq x \leq \frac{9}{4}\), a closed circle is drawn at 1 and \(\frac{9}{4}\) to show that these numbers are included in the solution.
The section between these endpoints is shaded to indicate all numbers within this range are part of the solution set.
Graphs make it easier to understand the scope of solutions and confirm that both inequalities overlap correctly in a compound inequality scenario.
To graph \(1 \leq x \leq \frac{9}{4}\), a closed circle is drawn at 1 and \(\frac{9}{4}\) to show that these numbers are included in the solution.
The section between these endpoints is shaded to indicate all numbers within this range are part of the solution set.
- Closed circles represent included endpoints (use square brackets in interval notation).
- Open circles represent excluded endpoints (use parentheses in interval notation).
Graphs make it easier to understand the scope of solutions and confirm that both inequalities overlap correctly in a compound inequality scenario.
Other exercises in this chapter
Problem 28
Factor difference of two squares. \(4 a^{2} b^{4}-9 d^{6}\)
View solution Problem 28
Solve each equation. \(|4 x-24.8|=32.4\)
View solution Problem 28
Solve each equation. $$ \frac{3}{2}(y+4)=\frac{20-y}{2} $$
View solution Problem 29
Determine whether the relation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds t
View solution