Question

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date
practice 18

1. graph the solutions to ( x geq 2 ) on the number line.
2. graph the solutions to ( p < 7 ) on the number line.
3. what are three solutions to the inequality in problem 2?
4. how many solutions to the inequality in problem 1 exist? explain.
5. write two inequalities that each represent the solutions graphed on the number line. use ( x ) as the variable.
6. adesh says that ( x > 3 ) has the same solutions as ( 3 > x ). yuna says that ( x > 3 ) has the same solutions as ( 3 < x ). which of the following statements is true?

a. only adesh is correct.
b. only yuna is correct.
c. both adesh and yuna are correct.
d. neither adesh nor yuna is correct.

Explanation:

Step1: Graph $x\geq2$ on number line

1. Locate 2 on the number line.
2. Draw a closed circle at 2 (since the inequality includes equality, $\geq$).
3. Shade all numbers to the right of 2 (including 2) to represent all values greater than or equal to 2.

Step2: Graph $p<7$ on number line

1. Locate 7 on the number line.
2. Draw an open circle at 7 (since the inequality does not include equality, $<$).
3. Shade all numbers to the left of 7 (not including 7) to represent all values less than 7.

Step3: Identify 3 solutions for $p<7$

Any real number less than 7 is a solution. Choose 3 examples:
$p=0$, $p=5$, $p=-3$

Step4: Count solutions for $p<7$

Inequalities of the form $p<7$ include all real numbers less than 7. There are infinitely many real numbers in this range.

Step5: Write inequalities for the given graph

The graph has a closed circle at -1 and shades all numbers to the left. This represents values less than or equal to -1. Two equivalent inequalities are:
$x\leq -1$ or $-1\geq x$

Step6: Evaluate Adesh and Yuna's claims

• Adesh claims $x\geq3$ is the same as $3\geq x$. $3\geq x$ is equivalent to $x\leq3$, which is the opposite of $x\geq3$, so Adesh is incorrect.
• Yuna claims $x\geq3$ is the same as $3\leq x$. Rearranging $3\leq x$ gives $x\geq3$, which matches the original inequality, so Yuna is correct.

Answer:

1. (Graph: Closed circle at 2, shade right)
2. (Graph: Open circle at 7, shade left)
3. $0$, $5$, $-3$ (any three numbers less than 7 are acceptable)
4. Infinitely many solutions. All real numbers less than 7 are solutions, and there is no limit to the number of real numbers in this set.
5. $x\leq -1$ and $-1\geq x$
6. B. Only Yuna is correct.