Problem 30 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ 7 x+6 \leq 48 \text { and }-4 x+3 \geq-21 $$

Step-by-Step Solution

Answer

The solution set is \[ x \leq 6 \]. Interval notation: \[ (-\infty, 6] \]

Step 1 - Solve the first inequality

Start with the inequality:\[ 7x + 6 \leq 48 \]Subtract 6 from both sides:\[ 7x \leq 42 \]Divide by 7:\[ x \leq 6 \]

Step 2 - Solve the second inequality

Start with the inequality:\[ -4x + 3 \geq -21 \]Subtract 3 from both sides:\[ -4x \geq -24 \]Divide by -4 (remember to reverse the inequality sign):\[ x \leq 6 \]

Step 3 - Find the intersection of solutions

For the inequalities to both be true, find the intersection of the solution sets from Steps 1 and 2. Both inequalities simplify to:\[ x \leq 6 \]

Step 4 - Write the interval notation

The solution set in interval notation is:\[ (-\infty, 6] \]

Step 5 - Graph the solution set

To graph \[ (-\infty, 6] \], draw a number line. Shade all the numbers to the left of 6, including 6 itself, which is represented by a closed circle at 6.

Key Concepts

inequality solving

Math inequalities can be solved similarly to equations, but they have an important difference. When solving an inequality, you may need to reverse the inequality sign under certain conditions.
For example, if you multiply or divide both sides of the inequality by a negative number, you must flip the sign.
It's crucial to pay attention to this while solving to get accurate results.

interval notation

Interval notation is a concise way to represent sets of numbers between two endpoints. It uses brackets to include or exclude endpoints:

• '[' or ']' means the number is included (less than or equal to).
• '(' or ')' means the number is excluded (strictly less than).

Consider our solved example. The solution to the inequality was \( x \leq 6 \). Here, all numbers less than or equal to 6 are part of the solution set. This set is written in interval notation as \((-\infty, 6] \). The \( -\infty \) indicates no lower bound, while 6 is included, shown by the closed bracket.

graphing inequalities

Graphing inequalities can give a visual representation of the solution set. Here's a simple way to do it:

• Draw a number line.
• Locate the number in the inequality (6 in our example) and place a point either open (if the number is not included) or closed (if included).
• Shade the region where the numbers are solutions to the inequality.

For \(( -\infty, 6] \), you'd place a closed circle at 6 and shade everything to the left, showing all values less than or equal to 6.

intersection of solutions

The intersection of solutions is important for compound inequalities. When you have multiple inequalities combined by 'and,' like in our example, you need to find where both conditions are true simultaneously. This intersection is the overlap of the solution sets of individual inequalities.
For instance, in the solutions to \( 7x + 6 \leq 48 \) and \( -4x + 3 \geq -21 \, \ x\leq 6 \) was the common solution. This overlapping region or 'intersection' becomes the final solution set you graph or write in interval notation.