Question

section 1.5 daily wo
solve by graphing.

1. ⅓x − 4 = −2x + 3
2. |x + 5| = 2x + 4
3. ¾x + 3 < −3
4. −|x| + 5 ≥ −1
5. −½x + 2 > |x − 1|
6. 2|x| + 3 ≤ −x + 6

Explanation:

---

Problem 1: $\frac{1}{3}x - 4 = -2x + 3$

Step1: Isolate x terms

$\frac{1}{3}x + 2x = 3 + 4$

Step2: Combine like terms

$\frac{1}{3}x + \frac{6}{3}x = 7$
$\frac{7}{3}x = 7$

Step3: Solve for x

$x = 7 \times \frac{3}{7}$
$x = 3$
---

Problem 2: $|x + 5| = 2x + 4$

Step1: Case 1: $x + 5 \geq 0$ (i.e., $x \geq -5$)

$x + 5 = 2x + 4$
$5 - 4 = 2x - x$
$x = 1$ (valid, since $1 \geq -5$)

Step2: Case 2: $x + 5 < 0$ (i.e., $x < -5$)

$-(x + 5) = 2x + 4$
$-x - 5 = 2x + 4$
$-5 - 4 = 2x + x$
$3x = -9$
$x = -3$ (invalid, since $-3
less -5$)
---

Problem 3: $\frac{3}{4}x + 3 < -3$

Step1: Isolate x term

$\frac{3}{4}x < -3 - 3$
$\frac{3}{4}x < -6$

Step2: Solve for x

$x < -6 \times \frac{4}{3}$
$x < -8$
---

Problem 4: $-|x| + 5 \geq -1$

Step1: Isolate absolute value

$-|x| \geq -1 - 5$
$-|x| \geq -6$

Step2: Reverse inequality (multiply by -1)

$|x| \leq 6$

Step3: Solve absolute value inequality

$-6 \leq x \leq 6$
---

Problem 5: $-\frac{1}{2}x + 2 > |x - 1|$

Step1: Case 1: $x - 1 \geq 0$ (i.e., $x \geq 1$)

$-\frac{1}{2}x + 2 > x - 1$
$2 + 1 > x + \frac{1}{2}x$
$3 > \frac{3}{2}x$
$x < 2$ (valid, since $1 \leq x < 2$)

Step2: Case 2: $x - 1 < 0$ (i.e., $x < 1$)

$-\frac{1}{2}x + 2 > -(x - 1)$
$-\frac{1}{2}x + 2 > -x + 1$
$-\frac{1}{2}x + x > 1 - 2$
$\frac{1}{2}x > -1$
$x > -2$ (valid, since $-2 < x < 1$)

Step3: Combine valid intervals

$-2 < x < 2$
---

Problem 6: $2|x| + 3 \leq -x + 6$

Step1: Isolate absolute value

$2|x| \leq -x + 6 - 3$
$2|x| \leq -x + 3$

Step2: Case 1: $x \geq 0$

$2x \leq -x + 3$
$2x + x \leq 3$
$3x \leq 3$
$x \leq 1$ (valid, since $0 \leq x \leq 1$)

Step3: Case 2: $x < 0$

$2(-x) \leq -x + 3$
$-2x \leq -x + 3$
$-2x + x \leq 3$
$-x \leq 3$
$x \geq -3$ (valid, since $-3 \leq x < 0$)

Step4: Combine valid intervals

$-3 \leq x \leq 1$
---

Answer:

1. $x = 3$
2. $x = 1$
3. $x < -8$
4. $-6 \leq x \leq 6$
5. $-2 < x < 2$
6. $-3 \leq x \leq 1$