Question

practice & problem solving
leveled practice in 6 and 7, fill in the boxes to solve each inequality. then graph the solutions.

1. (x + 5lt7)

(x + 5-squarecirc7-square)
(xcircsquare)
(number line from - 5 to 5)

1. (x - 4geq12)

(x - 4circsquaregeq12circsquare)
(xgeqsquare)
(number line from 8 to 16)

1. solve (10 + xgeq14). then graph the solution.

(empty number line)

1. solve (- 11geq x - 20). then graph the solution.

(empty number line)

1. check for reasonableness the maximum number of students in a classroom is 26. if there are 16 students signed up for the art class, how many more students can join the class without exceeding the maximum?
2. higher order thinking the inequality (x + cgt - 2.55) has the solution (xgt4.85) what is the value of (c)? how do you know?
3. represent and connect rina is climbing a mountain. she has reached base camp. write an inequality to find the maximum distance, (d), in feet that rina could climb between base camp and the peak.

(image of a mountain with peak 12,358 ft and base camp 9,695 ft)
5 - 3 solve inequalities using addition or subtraction

Explanation:

Problem 6: Solve \( x + 5 < 7 \)

Step 1: Subtract 5 from both sides

To isolate \( x \), we subtract 5 from both sides of the inequality. This is based on the subtraction property of inequalities, which states that if \( a < b \), then \( a - c < b - c \) for any real number \( c \).
\( x + 5 - 5 < 7 - 5 \)

Step 2: Simplify both sides

Simplifying the left - hand side, \( x+5 - 5=x \). Simplifying the right - hand side, \( 7 - 5 = 2 \). So we get \( x<2 \)

Step 1: Add 4 to both sides

To isolate \( x \), we use the addition property of inequalities. If \( a\geq b \), then \( a + c\geq b + c \) for any real number \( c \). So we add 4 to both sides of the inequality.
\( x - 4+4\geq12 + 4\)

Step 2: Simplify both sides

Simplifying the left - hand side, \( x-4 + 4=x \). Simplifying the right - hand side, \( 12 + 4=16 \). So we get \( x\geq16 \)

Step 1: Subtract 10 from both sides

Using the subtraction property of inequalities (if \( a\geq b \), then \( a - c\geq b - c \) for any real number \( c \)), we subtract 10 from both sides of the inequality \( 10 + x\geq14 \).
\( 10+x - 10\geq14 - 10\)

Step 2: Simplify both sides

Simplifying the left - hand side, \( 10 + x-10=x \). Simplifying the right - hand side, \( 14 - 10 = 4 \). So the solution is \( x\geq4 \)
To graph the solution:

• Draw a number line.
• Put a closed circle at 4 (because the inequality is \( \geq \), which includes 4) and shade the region to the right of 4 (since \( x\) is greater than or equal to 4).

Answer:

In the first two boxes (the ones we subtract) we put 5, the inequality sign is \( < \), and the last box is 2. So the filled - in steps are \( x + 5-5<7 - 5\) and \( x < 2\)

Problem 7: Solve \( x - 4\geq12 \)