Answer :
To determine the truth of each statement about the graph of the inequality [tex]\(y < \frac{2}{3}x + 1\)[/tex], let's analyze each statement in detail:
1. The slope of the line is 1:
The given inequality is [tex]\(y < \frac{2}{3}x + 1\)[/tex]. The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. In this case, the slope [tex]\(m\)[/tex] is [tex]\(\frac{2}{3}\)[/tex].
Therefore, the statement "The slope of the line is 1" is FALSE. The slope is actually [tex]\(\frac{2}{3}\)[/tex].
2. The line is solid:
The given inequality is [tex]\(y < \frac{2}{3}x + 1\)[/tex]. When the inequality does not include the equality (i.e., it is not [tex]\(y \leq \frac{2}{3}x + 1\)[/tex]), the boundary line is dashed, not solid.
Therefore, the statement "The line is solid" is FALSE. The line represents "less than" and is dashed.
3. The area below the line is shaded:
The inequality is [tex]\(y < \frac{2}{3}x + 1\)[/tex]. This indicates that the region where [tex]\(y\)[/tex] is less than the line [tex]\(\frac{2}{3}x + 1\)[/tex] is shaded, which is the area below the line.
Therefore, the statement "The area below the line is shaded" is TRUE.
4. A solution to the inequality is [tex]\((2,3)\)[/tex]:
To check if the point [tex]\((2, 3)\)[/tex] is a solution to the inequality [tex]\(y < \frac{2}{3}x + 1\)[/tex], we substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex] into the inequality:
[tex]\[ 3 < \frac{2}{3} \cdot 2 + 1 \][/tex]
[tex]\[ 3 < \frac{4}{3} + 1 \][/tex]
[tex]\[ 3 < \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \approx 2.333 \][/tex]
Since [tex]\(3\)[/tex] is not less than [tex]\(\approx 2.333\)[/tex], the point [tex]\((2, 3)\)[/tex] does not satisfy the inequality.
Therefore, the statement "A solution to the inequality is [tex]\((2, 3)\)[/tex]" is FALSE.
5. The [tex]\(x\)[/tex]-intercept of the boundary line is [tex]\(\left(-\frac{3}{2}, 0\right)\)[/tex]:
To find the [tex]\(x\)[/tex]-intercept, we set [tex]\(y = 0\)[/tex] in the equation of the line [tex]\(y = \frac{2}{3}x + 1\)[/tex]:
[tex]\[ 0 = \frac{2}{3}x + 1 \][/tex]
[tex]\[ \frac{2}{3}x = -1 \][/tex]
[tex]\[ x = -1 \div \frac{2}{3} = -1 \cdot \frac{3}{2} = -\frac{3}{2} \][/tex]
The [tex]\(x\)[/tex]-intercept is indeed [tex]\(\left(-\frac{3}{2}, 0\right)\)[/tex].
Therefore, the statement "The [tex]\(x\)[/tex]-intercept of the boundary line is [tex]\(\left(-\frac{3}{2}, 0\right)\)[/tex]" is TRUE.
In conclusion, the true statements about the graph of [tex]\(y < \frac{2}{3}x + 1\)[/tex] are:
- The area below the line is shaded.
- The [tex]\(x\)[/tex]-intercept of the boundary line is [tex]\(\left(-\frac{3}{2}, 0\right)\)[/tex].
1. The slope of the line is 1:
The given inequality is [tex]\(y < \frac{2}{3}x + 1\)[/tex]. The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. In this case, the slope [tex]\(m\)[/tex] is [tex]\(\frac{2}{3}\)[/tex].
Therefore, the statement "The slope of the line is 1" is FALSE. The slope is actually [tex]\(\frac{2}{3}\)[/tex].
2. The line is solid:
The given inequality is [tex]\(y < \frac{2}{3}x + 1\)[/tex]. When the inequality does not include the equality (i.e., it is not [tex]\(y \leq \frac{2}{3}x + 1\)[/tex]), the boundary line is dashed, not solid.
Therefore, the statement "The line is solid" is FALSE. The line represents "less than" and is dashed.
3. The area below the line is shaded:
The inequality is [tex]\(y < \frac{2}{3}x + 1\)[/tex]. This indicates that the region where [tex]\(y\)[/tex] is less than the line [tex]\(\frac{2}{3}x + 1\)[/tex] is shaded, which is the area below the line.
Therefore, the statement "The area below the line is shaded" is TRUE.
4. A solution to the inequality is [tex]\((2,3)\)[/tex]:
To check if the point [tex]\((2, 3)\)[/tex] is a solution to the inequality [tex]\(y < \frac{2}{3}x + 1\)[/tex], we substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex] into the inequality:
[tex]\[ 3 < \frac{2}{3} \cdot 2 + 1 \][/tex]
[tex]\[ 3 < \frac{4}{3} + 1 \][/tex]
[tex]\[ 3 < \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \approx 2.333 \][/tex]
Since [tex]\(3\)[/tex] is not less than [tex]\(\approx 2.333\)[/tex], the point [tex]\((2, 3)\)[/tex] does not satisfy the inequality.
Therefore, the statement "A solution to the inequality is [tex]\((2, 3)\)[/tex]" is FALSE.
5. The [tex]\(x\)[/tex]-intercept of the boundary line is [tex]\(\left(-\frac{3}{2}, 0\right)\)[/tex]:
To find the [tex]\(x\)[/tex]-intercept, we set [tex]\(y = 0\)[/tex] in the equation of the line [tex]\(y = \frac{2}{3}x + 1\)[/tex]:
[tex]\[ 0 = \frac{2}{3}x + 1 \][/tex]
[tex]\[ \frac{2}{3}x = -1 \][/tex]
[tex]\[ x = -1 \div \frac{2}{3} = -1 \cdot \frac{3}{2} = -\frac{3}{2} \][/tex]
The [tex]\(x\)[/tex]-intercept is indeed [tex]\(\left(-\frac{3}{2}, 0\right)\)[/tex].
Therefore, the statement "The [tex]\(x\)[/tex]-intercept of the boundary line is [tex]\(\left(-\frac{3}{2}, 0\right)\)[/tex]" is TRUE.
In conclusion, the true statements about the graph of [tex]\(y < \frac{2}{3}x + 1\)[/tex] are:
- The area below the line is shaded.
- The [tex]\(x\)[/tex]-intercept of the boundary line is [tex]\(\left(-\frac{3}{2}, 0\right)\)[/tex].
